The Democrat planned to attend a joint session of Congress on Tuesday, but Burris spokesman Jim O'Connor did not yet know with whom the senator might meet.
Fellow Illinois Sen. Dick Durbin wants to schedule a meeting with Burris "as soon as possible," spokesman Joe Shoemaker said. Durbin has not called for Burris to resign, saying he wants to talk to him in person before commenting further.
Senate Majority Leader Harry Reid, D-Nev., had no immediate plans to meet with Burris this week, said Reid spokesman Jim Manley.
Burris has resisted calls for his resignation, including from within his own party.
Burris testified in January before the Illinois House committee that recommended Gov. Rod Blagojevich's impeachment that he hadn't had contact with key Blagojevich staffers or offered anything in return for the seat. Blagojevich faces charges of trying to sell President Barack Obama's former Senate seat, though he denies wrongdoing.
But just over a week ago, Burris released an affidavit saying he had spoken to several Blagojevich advisers, including Robert Blagojevich, the former governor's brother and finance chairman, who Burris said called three times last fall asking for fundraising help.
He changed his story again last week when he admitted trying, unsuccessfully, to raise money for Blagojevich.
Now Burris, the nation's only black senator, finds himself caught up in a political firestorm.
Illinois lawmakers have asked local prosecutors to look into perjury charges, and a preliminary U.S. Senate Ethics Committee inquiry is under way. Even the White House said last week that Burris should take the weekend to consider his future.
Sen. Arlen Specter, R-Pennsylvania, said Monday that it is up to Burris to decide whether to resign.
"If there is an issue of perjury, then Senator Burris has the right to defend himself in any proceeding on that subject," Specter said in a statement.
If Burris decides to stay, he could find it difficult to recruit a staff and likely will be isolated politically, said Kent Redfield, a political science professor at the University of Illinois in Springfield.
"The national Democrats needed his vote, but they found that he hung them out to dry," said Redfield, adding he doubts the ethics panel will do anything to Burris. "The Democrats are not going to go out of their way to make him front and center in terms carrying legislation.
"They're going to avoid anything that might draw attention to him."
Meanwhile, Republican political adviser Karl Rove said the Illinois GOP can win back state offices in the wake of the impeachment of Blagojevich and the controversy around Burris.
Rove made his assertion to several hundred people at a party fundraiser late Monday in Schaumburg. However, the political adviser to former President George W. Bush told Republicans they can only win if they don't rip the party apart in next year's primaries.