Burris, also a Democrat, was appointed by disgraced former Gov.
Rod Blagojevich, who was impeached and driven from office after he was charged in a federal criminal complaint of trying to sell the Senate seat.
Burris repeatedly changed his story about how he was appointed.
He is facing calls for his resignation after he admitted trying to raise money for Blagojevich. Burris has said he did nothing wrong.
Emerging from the hour-long private meeting with Durbin, Burris looked a bit shaken and inexplicably said he was under orders not to comment, other than to say the session was a "great discussion."
Burris has faced intense pressure from all quarters, from politicians to home state newspapers clamoring for him to step down. Illinois Gov. Pat Quinn said last week Burris should resign for the good of the state, arguing the controversy surrounding his appointment has cast a shadow over his service in the Senate.
In their meeting, Durbin said he told Burris that support among other Democrats was eroding because of Burris' shifting account of whether he tried to raise money for Blagojevich.
And in the careful language of the Senate, Durbin said he made clear that if Burris tried to run for the seat next year, he would not have much - if any - support from Senate Democrats. Consistent with a Senate appointment, Burris would have to win the seat outright next year.
"I asked him if he would be a candidate in 2010 and he said he had not made up his mind," Durbin said. "I told him I thought it would be extremely difficult for him to be successful in a primary or a general election under the circumstances."
The conversation followed Burris' exchange earlier in the day with Senate Majority Leader Harry Reid, D-Nev., on the Senate floor.
"I said, 'How was your break?' He said, 'Fine. How was yours?"' Reid told reporters afterward. "I said, 'Fine.' OK?"
Asked whether Reid thinks Burris should resign, spokesman Jim Manley said: "That is for him to decide."
The chilliness that greeted Burris in Washington came after outright demands for his resignation in Illinois.
Burris testified in January before the Illinois House committee that recommended 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 Obama's former Senate seat, though he denies wrongdoing.
But just more than 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.
Illinois lawmakers have asked local prosecutors to look into perjury charges, and a preliminary Senate Ethics Committee inquiry is under way. Even the White House said last week that Burris should take the weekend to consider his future.
Burris has denied any wrongdoing. Attorney Timothy W. Wright III said Tuesday that he was working on a document that would prove Burris didn't lie about the circumstances of his appointment.
