Beyond that, negotiators remain far apart on the number of beds for families detained at the border with Mexico. There currently are enough funds allocated for 40,520 detention beds, under the control of Immigration and Customs Enforcement. Democrats asked to limit the number of beds to 35,520 for the remainder of the fiscal year, and to increase funds for alternatives to the practice of family detention. Republicans want substantially more beds.

Disputes like that were part of why Democratic aides on Friday played down the likelihood of a quick deal. The Democrats’ last official offer on funding for barriers was a little more than $500 million — although privately they acknowledge they are likely to go higher.

Regardless of where the talks land, the funding under discussion is another significant compromise for Mr. Trump as he struggles to fulfill his campaign promise that he would build a wall at the southwestern border. In December, he publicly dismissed a proposal for $2.5 billion as insufficient, undercutting Vice President Mike Pence, who had floated the prospect by Democrats.

Mr. Leahy, a member of the bipartisan House-Senate committee negotiating the deal, declined to elaborate on any details, citing the need to keep the continuing talks private.

He said that he, along with staff and other members of the 17-member committee, planned to stay in Washington this weekend to negotiate the final details and prepare to sign off on it on Monday. Because of procedural rules in the House, it is seen as the last possible day for lawmakers to unveil a deal that would prevent a government shutdown before the Feb. 15 deadline.

Over the past several days, negotiators have begun discussing the potential placement of new barriers — some of it likely to be bollard fencing, made of sturdy, steel slats — in specific locations on the border that have seen significant increases in illegal crossings, according to two Republican aides.

“The walls we saw were really heavy fences,” said Representative Kay Granger of Texas, the ranking Republican on the House Appropriations Committee, describing a trip to the border she took this week. She told reporters that after speaking with Border Patrol agents, she would support bollard fencing on sections of the border.