WASHINGTON — The fate of President Trump’s $5.7 billion demand for a border wall is now in the hands of a 17-member bipartisan panel that includes some of the most senior members of Congress and, perhaps more tellingly, lacks the most vocal immigration hard-liners on Capitol Hill.

Under the agreement Mr. Trump reached last week with congressional Democrats, a committee of Republican and Democratic lawmakers from both chambers — known as a conference committee — has until Feb. 15 to come up with a border security package.

During the 35-day shutdown, many Americans accused Mr. Trump and Congress of acting like toddlers, with Mr. Trump insisting that he had to have the wall, and Democratic leaders insisting that they would not give him a penny for it.

Now, a group less dominated by ideology will be in charge.

One conference committee member, Senator Jon Tester, Democrat of Montana, departed from Democratic orthodoxy on Monday when he told reporters that he would be in favor of including some money for a wall in the border security package. Late last year, Mr. Tester and other Democrats on the Appropriations Committee voted for a spending bill that included $1.6 billion for 65 miles of fencing along the border.

“What I hope can happen is we end up with a set of broad-based measures to secure the border, including manpower, wall, technology,” Mr. Tester said. Noting that physical barriers already exist along roughly 600 miles of the border, he added, “I think to take that off the table entirely — this is my opinion, O.K.? — would be not the best direction to go.”

The conference panel’s members, appointed by House and Senate leaders, are drawn exclusively from the appropriations committees in each chamber, which have a long history of working in a bipartisan way. (Senator John McCain was fond of saying, “There are Republicans, there are Democrats, and then there are appropriators.”) The first meeting will be Wednesday afternoon.

With nine Democrats and eight Republicans, the panel traverses the philosophical spectrum: Representative Barbara Lee, Democrat of California, is an ardent liberal critic of Mr. Trump’s immigration policies, as is Senator Richard J. Durbin, Democrat of Illinois.

 Representative Tom Graves, Republican of Georgia, has co-sponsored legislation to provide $23.4 billion in wall construction money, while Representative Steven M. Palazzo, Republican of Mississippi, has backed a bill that would allow Americans to purchase bonds to help finance a wall. But neither is closely identified with that cause.

And the panel includes members of both parties, beyond Mr. Tester, who have been calling for compromise.

“This is much more than just a wall,” another Republican member, Representative Chuck Fleischmann of Tennessee, told Fox Business Network last week, in urging an end to the shutdown. “We’ve got to look at border security where there are areas where a wall won’t work; we’ve got to secure it that way. But yes, I think we can sit down and come up with something. The American people want a reasonable compromise that will work for all.”

Mr. Durbin helped write the Development, Relief, and Education for Alien Minors (Dream) Act in 2001, which would offer a path to citizenship for young immigrants who were brought illegally to the country as children. Members of both parties have suggested some version of the Dream Act could be exchanged for some wall funding.

Conference committees, once an elemental part of Congress, occur when the House and Senate have to reconcile two bills. But in the recent past, much of the reconciliation has been done by leadership. Genuine conference committees have become so rare that many lawmakers have never been on one that worked as intended.

The border security panel conference includes several Democrats from border states, including Ms. Lee; Representative Lucille Roybal-Allard of California; Representative Henry Cuellar of Texas; and Representative Pete Aguilar of California.

It will be led by the top Democrat and Republican appropriators in both chambers: Representatives Nita M. Lowey, Democrat of New York, and Kay Granger, Republican of Texas; and Senators Richard C. Shelby, Republican of Alabama, and Patrick J. Leahy, Democrat of Vermont.

Together, they have more than 100 years of legislative experience — Mr. Leahy alone has been in the Senate for 43 years — and are senior enough to remember the days of real conference committees, when lawmakers worked across party lines to hash out deals.

“In my judgment, the next three weeks will be one of the most important work periods in recent history — in my job here,” said Ms. Lowey, who was elected to the House in 1988 and is the first woman to lead its Appropriations Committee. She will be the chairwoman of the conference panel; Mr. Shelby will be its vice chairman.

“I’ve been an appropriator for a long time,” she added. “I’m always willing to work across the aisle and find compromise.”

What that compromise might look like — and whether it will satisfy Mr. Trump — is unclear.

At least one Trump ally on Capitol Hill — Senator Lindsey Graham, Republican of South Carolina — is suggesting that Mr. Trump go ahead and declare a national emergency to build the wall, without congressional support.

The Republicans on the panel are Representatives Granger, Fleischmann, Graves and Palazzo; Senators Shelby, John Hoeven of North Dakota, Shelley Moore Capito of West Virginia and Roy Blunt of Missouri.

The Democrats are Representatives Lowey, Roybal-Allard, Lee, Cuellar, Aguilar and David Price of North Carolina; Senators Durbin, Leahy and Tester. At Wednesday’s meeting, each member will be able to give an opening statement.

“We’ve got to come together; the question is, ‘Will we?”’ said Mr. Shelby. Asked if he was optimistic, he said, “Optimistic? I’m cautious, guarded, anxious.”