WGN-TV, Chicago, reported Brady says he won't concede with about 1 percent of the votes still to be counted, a process he says could take a month, the TV station said.
"It's important that we let this process move forward in an organized way and allow all the ballots be counted," Brady said. "As anxious as we all are, we're all going to have to be patient."
The Quinn campaign expressed measured confidence Quinn will eventually be declared the winner.
"The ballots left to be counted appear mostly to come from Cook County, where the governor held a large margin over Sen. Brady," Quinn campaign spokeswoman Mica Matsoff said in a statement. "We expect to hold our lead, and may increase it."
WGN said Cook County and Chicago election officials say they will not count the 15,000 outstanding absentee ballots until Thursday at the earliest as they work to make sure the ballots are valid.
Republican Rep. Mark Kirk won a tight race for the Illinois U.S. Senate seat formerly held by President Barack Obama.
Kirk beat state Treasurer Alexi Giannoulias by 8,000 votes out of 3.39 million votes cast. Giannoulias struggled to distance himself from the collapse of his family's Broadway Bank while Kirk had to explain a series of resume embellishments.
"I want to make sure that every vote is counted," said Quinn, who stepped up from lieutenant governor after Rod Blagojevich was impeached. "The people have won, and I believe we have won. We know there are more votes to be counted, but we are ... ahead."
Quinn has struggled with the state's fiscal crisis, sparring with Brady over taxes and spending cuts, the Chicago Tribune noted.
Three Democratic house incumbents -- Bill Foster, Debbie Halvorson and Phil Hare -- all fell victim to an anti-Democratic tide and lost to their Republican opponents, the Tribune said.