The Chicago Sun-Times and the Chicago Tribune reported Burris will make the decision official Friday.
The Sun-Times said Burris decided against a run because he was having trouble raising campaign funds. Papers to be filed next week with the Federal Election Commission are expected to show he has managed to garner only $20,000 in donations, the Sun-Times said.
Burris, who was appointed by impeached Gov. Rod Blagojevich, who is under federal indictment, took the seat formerly held by President Barack Obama. Among the charges against Blagojevich is he allegedly tried to sell Obama's seat to the highest bidder.
Word of Burris' decision follows by a day an announcement by Illinois Attorney General Lisa Madigan that she would not seek the Democratic nomination for Burris' seat, though she was seen as the front-runner. Her announcement was followed by one by Rep. Mark Kirk, R-Ill., that he would seek the Republican nod.
Burris was the first black politician elected to statewide office in Illinois and served as comptroller as well as attorney general -- the office currently held by Madigan.