WASHINGTON — When it became clear that Representative Nita M. Lowey, Democrat of New York, would lead the House Appropriations Committee, she received a cherrywood gavel, her name and new title engraved in silver.

It was a gift from Representative Kay Granger, Republican of Texas, a longtime committee colleague.

“It’s a lady’s gavel,” Ms. Granger said this past week, her own engraved red gavel by her side. “Powerful.”

And when Ms. Granger secured her position as ranking member, Ms. Lowey was the first to call to congratulate her.

Because of their committee ranks, both women represent House party leaders on the bipartisan panel of lawmakers negotiating a compromise on border security, and their unlikely partnership — Ms. Lowey, 81, is an ardent liberal, and Ms. Granger, 76, is a fierce conservative — is one reason their colleagues believe they can reach an agreement before government funding again lapses on Friday.