Two people briefed on the discussions said Mr. Trump had directly pressed Mr. Pence to find an alternative to certifying Mr. Biden’s win, such as preventing him from having 270 electoral votes and letting the election be thrown to the House to decide.

In Georgia on Monday night at a rally for Senators David Perdue and Kelly Loeffler, Mr. Trump openly pressured the vice president, saying, “I hope Mike Pence comes through for us, I have to tell you.” He added, “Of course, if he doesn’t come through, I won’t like him as much,” before saying that he really likes Mr. Pence.

On Monday, after Mr. Pence returned from Georgia, the vice president and Mr. Trump were expected to hear a last-minute pitch at the White House from John Eastman, another Trump lawyer. Mr. Pence also met with Senate parliamentarians for hours on Sunday to prepare himself and the president for what he would say while on the Senate floor.