CHARLESTON, W.V. — Senator Shelley Moore Capito, Republican of West Virginia, was juggling visits across her state and preparations to tee up a potential infrastructure deal when she learned that the top Senate Republican had described himself as “100 percent focused” on stopping President Biden’s agenda.

The comment from Senator Mitch McConnell of Kentucky, delivered in his home state, was a harsh reminder for Ms. Capito of the tricky politics she was navigating as the Republican responsible for figuring out if there is a bipartisan compromise to be had with Mr. Biden on a major infrastructure and public works plan.

Privately, Ms. Capito said in an interview last week at a construction site, wedged in between trips to meet with constituents and business owners, Mr. McConnell is “telling me to move forward, he’s telling me to negotiate in good faith.” But his public remark “did sound a little…,” she trailed off with a chuckle.

“I thought, now I’m going to go into the president and go, ‘Well, here we are to negotiate!’”

That is precisely what Ms. Capito plans to do on Thursday, when she is scheduled to lead a group of a half-dozen Republicans in a meeting with Mr. Biden at the White House to discuss the possibilities for an infrastructure compromise. The stakes are high and the odds long that a deal can be reached given Mr. McConnell’s approach — he has tried to walk back his remark and called for a “consultative process” on the issue — and the vast gulf between the two parties on what constitutes infrastructure, how much should be invested in it and how to finance it.

But if a bipartisan deal is to be struck on a substantial infrastructure package this year, Ms. Capito, 67, a recently re-elected second-term senator from West Virginia and a close ally of Mr. McConnell’s who describes herself as a “solid conservative vote,” is likely to be leading it.

Mr. Biden has proposed a $4 trillion plan, including a $2.3 trillion initiative that includes projects like roads and bridges that have traditionally dominated infrastructure packages and a huge expansion of safety net programs that he has labeled critical “human infrastructure” initiatives — all paid for with tax increases on corporations and high earners. Ms. Capito has drafted a blueprint for a counteroffer that would devote a small fraction of that amount — $568 billion — with little detail about how it would be financed.

Still Ms. Capito, the daughter of a former West Virginia governor and the first woman to represent the state in the Senate, is doggedly pursuing a deal. She has spoken personally with Mr. Biden in recent days and dispatched her staff to talk to White House aides about reconciling her framework with the president’s.

During a private phone call on the subject recently, Mr. Biden, she said, “was very respectful of what my position might be.” He did not back down from his hulking proposal either, she added.

“The first indication that Lucy is going to pull the football is if she quits talking to you, and we’re still talking daily,” Ms. Capito said. “Maybe by working together, and accomplishing something together, we get a mutual win here — particularly a win for the country.”

For Ms. Capito, the quest for an infrastructure deal is rooted deeply in her state and her family’s legacy. Ms. Capito grew up here, spending time hiking, picnicking and traveling over bridges like the New River Gorge Bridge in southern West Virginia, which was granted national park protections through legislation she helped write. It was built in part during the tenure of her father, Arch Alfred Moore Jr., who served three terms as governor before being sent to prison for a corruption scheme.

As she stood underneath the dark russet beams of the bridge on a recent rainy Wednesday afternoon, hundreds of feet above churning river water, Ms. Capito pointed out how the steel arch span — in reducing 40 minutes of driving over winding roads to less than a minute — had bolstered the economies for rural communities and drawn hordes of architecture enthusiasts and BASE jumpers.

“This is a lifeblood kind of infrastructure for these communities,” Ms. Capito said, the steel railings of the catwalk shuddering as traffic rumbled over the bridge above her. “That part of the infrastructure bill — this sort of demonstrates that, and in an important way.”

In one of the nation’s poorest states where dozens of buildings, roads and projects bear the name of Senator Robert E. Byrd, in recognition of the $10 billion in federal dollars he funneled their way, Ms. Capito said her constituents expected her to deliver on their needs and she believed that “government has a place to help you overcome the challenges.”

That approach has sometimes put her at odds with her own party. Sitting in her kitchen in Charleston, Ms. Capito likened her trips to the grocery store to town-hall-style meetings, where neighbors and shoppers stop her to ask about Washington, offer their thoughts on the latest issues and either complain about or compliment her work. (“I’ll pick up 10 things,” her husband, Charlie Capito, interjected, “and she’ll have gone from the peppers to celery.”)

“When tensions run high, it’s probably not the best — it’s not the most relaxing trip to the grocery store,” she said. During the health care debate in 2017, when she broke with her party and refused to support a repeal of the Affordable Care Act without a replacement, “I did get slammed in the pepper section one time.”

Infrastructure could give Ms. Capito an opportunity for a much more broadly popular position: targeting aid directly to her state. Her plan calls for devoting $299 billion to roads and bridges, although only about a third of the blueprint is new spending, with the rest repurposed from existing programs.

“She has truly got the state’s and the nation’s best interest at heart — that’s where she’s going,” said Byrd White III, the state’s secretary of transportation, adding Ms. Capito was not a “‘Hooray for me’ person.” “She’s got some pretty in-depth knowledge about what’s going on.”

In West Virginia, much of the infrastructure has deteriorated because of the challenges of the state’s topography, a lack of upkeep and little guarantee of reliable federal aid. In some parts of the state, it is difficult to get the broadband services needed to support students and parents working from home during the pandemic.

The difficulty of maintaining roads and bridges that wind through mountains, coupled with a declining lack of revenue in part from the state’s beleaguered coal industry, contributed to a D rating from the state’s chapter of the American Society of Civil Engineers in December. (The country overall received a C-.)

“I drove over six bridges to take my kids to school — statistically speaking, one of those are structurally deficient,” said Tabitha Lafferre, an assistant professor of engineering technology at Fairmont State University and one of the lead engineers on the report card. “That bill is now overdue as far as investment and, you know, maintenance and backlogs and things like that.”

In a series of interviews across her state last week, Ms. Capito acknowledged steep challenges in reaching a deal with Mr. Biden to deliver such legislation.

Mr. McConnell has repeatedly raised $600 billion as an acceptable price tag, and Republicans have refused to consider tax increases that would reverse the deep cuts they pushed through as part of the 2017 tax law. Several Democrats, for their part, have dismissed Ms. Capito’s plan as paltry given the nation’s infrastructure needs.

After muscling nearly $1.9 trillion in pandemic relief through Congress without any Republican votes, Mr. Biden and White House officials have insisted that they want Republican support for a substantial investment in infrastructure. While introducing both pieces of Mr. Biden’s $4 trillion economic package, top cabinet secretaries and administration officials have conducted hundreds of calls with lawmakers and legislative staff to walk them through the plans, in addition to holding dozens of meetings and briefings.

“The president and White House have been in close contact with Senator Capito, and we appreciate the good will she’s shown throughout the process,” Louisa Terrell, the director of the White House legislative affairs office, said in a statement. “We hope to continue that conversation this week with the senator, and with any of her Republican colleagues who are willing to negotiate in good faith on a path forward.”

Moderates including Senator Joe Manchin III, Democrat of West Virginia, are pushing to find a compromise and have applauded, if not fully endorsed, Ms. Capito’s efforts. She readily concedes that her efforts could fall flat without a compromise that Democrats would be willing to accept — or if Republicans refuse to coalesce behind one, leaving Democrats to conclude it would be futile to winnow down the size of their plan in search of a bipartisan deal.

But she appears determined to try.

“At the end of the day I think most of us — we want to do things, rather than just figure out the best way to make the other person look bad,” Ms. Capito said. “We’ve spent enough time doing that. I think it’s time to seize our openings.”