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.