"To step aside and resign is, I think, a heroic act, and I ask Roland to do that," Quinn said at a news conference.
In the event Burris does resign, Quinn said lawmakers should quickly pass legislation to fill any Senate vacancy by special election, rather than gubernatorial appointment.
"At no time should our state go without full and fair representation in the United States Senate," Quinn said. He declined to say who he might temporarily appoint if Burris resigns.
Burris has given no indication of heeding the many calls for him to resign, including from other Democratic lawmakers. He was on a listening tour of the state Friday and is refusing to speak publicly about the controversy. Messages seeking comment on Quinn's statements were not immediately returned.
Burris, a former Illinois attorney general, testified before the Illinois House committee that recommended Blagojevich's impeachment in January that he hadn't had contact with key Blagojevich staffers or offered anything in return for the Senate seat vacated by President Barack Obama.
Last weekend, however, 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. Burris, a former state attorney general, changed his story again this week when he admitted trying, unsuccessfully, to raise money for Blagojevich.
Illinois lawmakers have asked local prosecutors to look into perjury charges, and a preliminary U.S. Senate Ethics Committee inquiry is under way. Burris denies lying under oath.