In a press conference this afternoon, Burris said he was pleased with the meeting and denied having any information about Blagojevich's wiretapping scandal.
Responding to questions on whether supporters had actually made calls to Blagojevich on his behalf, Burris laughingly responded, "If they did, it was certainly no 'pay to play' involved, because I don't have no money."
When asked whether the senators were offering him a deal to be senator in return for his promise not to run in the 2010 re-election, Burris denied knowing anything about that.
"I want to know where this information comes from. This wasn't even on their radar screen; they haven't even bought it up," he said. "I got these rumors about conditions. ... It didn't come up."
Reid and Durbin emerged from the 45-minute conference with Burris to say that the meeting was "positive" and that Burris is a fine fellow, but he is not yet eligible for the Illinois Senate seat.
"Roland Burris to me appears to be candid and forthright. ... He is not trying to avoid any responsibility or hide anything. ... We don't have a problem with him as an individual," Reid said. "[But] we want to do what we can do to make sure that everything has been done in the Blagojevich hours is transparent."
The senators said Burris had prepared an affidavit, and that the process is likely to be expedited once he has appeared before Blagojevich's investigation committee and received all the necessary approvals. The Illinois Supreme Court is now considering whether to compel the secretary of state to sign, and Durbin said he called the court to urge a quick decision and was assured a decision would be reached today or Thursday.
"We believe that the certificate by the secretary of state is vital to comply with Senate rules. We'll reassess where we are when that's done," Reid said.
Reid said that, ultimately, the full Senate would need to vote on whether to seat Burris.
The two also brought up the topic of race, emphasizing that Burris agreed the issue was not about race. If Burris joins the Senate, he will be its only African American senator.
Burris is set to testify before the House impeachment committee in Illinois Thursday, but clearing the other hurdle may not be so easy. The Illinois secretary of state has filed objections with the Illinois Supreme Court to Burris' motion.
ABC News' George Stephanopoulos reported earlier that the deal could involve giving the seat to Burris if he agrees not to run for reelection in 2010 because of concerns that he couldn't win that race for the Democrats.
But Burris denied there was a deal, saying he had never heard of one.
According to Stephanopoulos, the lieutenant governor of Illinois has indicated that he could go along with that deal, but there are no signs that Burris would accept the terms.
Even if he does accept, it remains to be seen whether Burris would keep the promise not to run in 2010.
Responding to questions on whether he would run in 2010, Burress said, "Well, now let me get my Senate legs under me, and get in and raise some money for all this stuff we've been doing, and get in and get settled and learn where the bathrooms are."
Reid and other Senate Democrats had earlier vowed not to seat anyone appointed by Blagojevich.