The clamour began after an affidavit was released over the weekend showing Burris was contacted by ousted Gov. Rod Blagojevich's brother about making a campaign contribution.
"I can't believe anything that comes out of Mr. Burris at this point," state Rep. Jim Durkin said. "I think it would be in the best interest of the state if he resigned because I don't think the state can stand this anymore."
Durkin and House Republican Leader Tom Cross want an investigation of Burris for possible perjury.
The affidavit, filed with the Illinois House Impeachment Committee on Feb. 5, contradicts statements he made last month in front of the committee, before being sworn in as senator on Jan. 15.
The discrepancy could mean he perjured himself.
On Monday, Burris said the affidavit was voluntary and not the result of contact from federal agents investigating the former governor. He said any suggestion otherwise is "absolutely, positively not true."
He also insisted the affidavit doesn't go back on previous sworn statements to lawmakers who recommended Blagojevich's impeachment.
On Sunday, an adamant Burris told reporters at a combative press conference in Chicago that he hadn't done anything wrong and never misled anyone.
"I've always conducted myself with honor and integrity," said Burris, 71. "At no time did I ever make any inconsistent statement."
Burris' office released the affidavit after its existence was first reported by the Chicago Sun-Times. The U.S. senator said Sunday he spoke to U.S. Senate Majority Leader Harry Reid and senior Illinois Sen. Dick Durbin and "they understand what's going on."
Reid spokesman Jim Manley confirmed Burris told Reid about the affidavit Friday.
"Clearly it would have been better if Senator Burris had provided this information when he first testified," Manley said. "Senator Reid is reviewing the affidavit and will await any action by Illinois legislative leaders after they review the matter."
Durbin spokeswoman Christina Angarola said Burris told Durbin about the affidavit on Friday, but didn't provide a copy.
Gov. Pat Quinn, who advanced to the governor's mansion after Blagojevich was ousted over corruption allegations last month, also called on Burris to explain the contradiction.
"My opinion is that he owes the people of Illinois a complete explanation," Illinois Gov. Pat Quinn said, according to Quinn spokesman Bob Reed.
According to the affidavit, Blagojevich's brother, Robert Blagojevich, called Burris three times — once in October and twice after the November election — to seek his fundraising assistance.
The disclosure reflects a major omission from Burris' testimony in January when an Illinois House impeachment committee specifically asked if he had ever spoken to Robert Blagojevich or other aides to the now-deposed governor about the seat vacated by President Barack Obama.
Burris explained Sunday that he never got a chance to answer a direct question about Blagojevich's brother, and submitted the Feb. 4 affidavit to clarify.
However, transcripts of Burris' impeachment committee testimony show he had opportunities to provide a full response to Illinois legislators. In one instance, when asked directly about speaking to Robert Blagojevich and other associates of the former governor, Burris consulted with his attorney before responding.
Robert Blagojevich's attorney has said that his client believes one of the conversations was recorded by the FBI.
Burris said Sunday that he told Robert Blagojevich he would not raise money because it would look like he was trying to win favor from the governor for his appointment.
"I did not donate one single dollar nor did I raise any money or promise favors of any kind to the governor," he said.
It's not clear what action state legislators could now take against Burris, said Northwestern University law professor and former Illinois Comptroller Dawn Clark Netsch.
"I'm not aware that anything quite like this has happened in any state before," she said.
Based on federal law, the state Senate could argue that Burris was a temporary appointment, then pass a bill calling for a special election to name a permanent senator, Netsch said.
But Quinn's hands may be tied."I don't see anything that the current governor could do, except to ask for legislation to ask for a special election," she said.
Durkin and Cross said Sunday that they think a special election would be the best course.