By Friday, Burris eluded reporters and only repeated pleas through his spokesman to "stop the rush to judgment." The junior senator scheduled private visits to a veterans medical center and naval training center north of Chicago but did not speak publicly.
Quinn praised Burris, a former Illinois attorney general, as an honorable man and said Friday it would be a "heroic act" for him to put the interests of the state and his constituents ahead of his own.
"It takes a great deal of fortitude and courage to do that and I think people will recognize that," Quinn said. "It's never easy to step aside and resign from anything in life, especially something as important as United States Senate."
Burris, meanwhile, gave no indication he would heed the calls to give up the coveted seat.
"Like he said before, he's asked the public and officials to stop the rush to judgment and to allow all of the facts to come out," spokesman Jim O'Connor said.
Burris testified before the Illinois House committee that recommended Blagojevich's impeachment in January that he hadn't contacted key Blagojevich staffers or offered anything in return for the Senate seat vacated by President Barack Obama. Blagojevich faces federal charges that he tried to sell the seat.
Last weekend, however, Burris released an affidavit saying he had spoken to several Blagojevich advisers, including the former governor's brother and finance chairman, who Burris said called three times last fall asking for fundraising help.
Burris this week acknowledged trying, unsuccessfully, to raise money for Blagojevich.
Illinois lawmakers have asked prosecutors to look into perjury charges, and a preliminary U.S. Senate Ethics Committee inquiry is under way. State Rep. Jack Franks introduced a resolution urging the Illinois Senate Ethics Committee to expel Burris.
In the event Burris resigns or is expelled, Quinn has implored lawmakers to quickly pass legislation, already introduced by Franks, to fill any Senate vacancy by special election rather than gubernatorial appointment.
Quinn wants the Illinois Legislature to allow him to name a temporary replacement to a vacant seat until a special election could be held. He declined to say who he might appoint if Burris steps aside.
"At no time should our state go without full and fair representation in the United States Senate," Quinn said.
Illinois Republicans don't want to wait for Burris to quit and are calling for a May 26 election even if he doesn't resign.
The GOP argues Burris' appointment is only good until the next election, so they want to move up the next one and let voters decide - effectively kicking him out of he doesn't win.
Illinois lawmakers had considered stripping Blagojevich of his Senate appointment powers after his Dec. 9 arrest on federal corruption charges, but they couldn't agree on legislation. Blagojevich shocked everyone by naming Burris to the seat Dec. 30.
Rikeesha Phelon, a spokeswoman for Illinois Senate President John Cullerton, said the Democrat was "open" to special elections but has not committed to a proposal. Cullerton has not called for Burris' resignation, saying he wants investigations to play out.
Democratic House Speaker Michael Madigan's spokesman, Steve Brown, said they're reviewing proposed legislation, including where special election funding would come from.
GOP state Sen. Matt Murphy has estimated a special election could cost $5 million to $25 million.
Quinn said money shouldn't hinder voters from picking their next senator.
"In a Democracy, you don't say we don't have the voice of the people heard because it costs too much," he said.
Quinn's pleas for Burris to resign joined numerous others in recent days, including Illinois Treasurer Alexi Giannoulias and other elected officials. The White House did not go so far, but said Burris must explain conflicting statements.
"I think it might be important for Senator Burris to take some time this weekend to either correct what has been said and - and certainly think of what lays in his future," White House press secretary Robert Gibbs added.
Also Friday, Burris' interim chief of staff left to return to work as a senior adviser for Senate Majority Leader Harry Reid. Darrel Thompson said he was temporarily detailed three weeks ago to serve Burris, but that the job ended Friday.
It was not immediately clear who would replace Thompson.