"If it doesn't die down, I think the [Senate Republican] caucus will be forced give a vote of confidence or no confidence in him," the aide said.
Lott left Washington early this week; his office has declined to reveal his whereabouts.
Officials also pressed the effort to distance the White House from Lott's comments.
Without mentioning Lott by name, White House political director Ken Mehlman said Bush has often commented that no political party can be "a great national party that is not winning a greater percentage" of the minority vote.
Democratic presidential candidates routinely have won 80% to 90% of the black vote for decades.
Mehlman, speaking at a political conference in Washington, said the best strategy for the GOP to improve its support among blacks "is to produce results" in education reform and other social programs.
Lott, 61, has led Senate Republicans as majority leader or minority leader since Sen. Bob Dole of Kansas stepped down in 1996 to run for president. He remained in charge of the Senate GOP even after Sen. James M. Jeffords of Vermont last year quit the GOP and became an independent, handing control of the Senate to Democrats.
Lott was riding high after November's elections returned control of the Senate to the Republicans and his GOP colleagues again selected him as majority leader.
But his political life took a dramatic turn last week when, at Thurmond's 100th birthday party, Lott noted that Mississippi backed Thurmond's segregationist Dixiecrat presidential candidacy in 1948.
"We're proud of it," Lott said at the gathering. "And if the rest of the country had followed our lead, we wouldn't have had all these problems over all these years, either."
Philadelphia Mayor John Street, a Democrat who attended Bush's speech, refrained from saying whether Lott should resign his leadership job.
But he said Lott's remarks "are more than just regrettable.... I'm not sure a simple apology is good enough. It's inappropriate for something like this to come from someone in a leadership position."
Times staff writers Don Frederick and Maura Reynolds contributed to this report.