Terms of the agreement, which is pending McDonald passing a physical exam, are not yet known.
McDonald was released by the San Francisco 49ers in December after law enforcement officials in San Jose, California, said he was under investigation on suspicion of sexual assault. McDonald hasn't been charged in that case, which remains open.
Bears general manager Ryan Pace told reporters at the owners meetings in Arizona that, because there were character issues with McDonald to consider, Bears chairman George McCaskey weighed in on the decision before the team went through with the move.
McDonald paid for his own ticket to travel to Chicago and met with McCaskey for two hours, Pace said. George McCaskey also consulted with Virginia McCaskey and received her blessing before signing off on McDonald's addition.
"I was impressed with how sincere he was and how motivated he is," McCaskey told reporters Tuesday night at the owners meetings in Phoenix, referring to the meeting he had with McDonald. "He understands, I think, that he could have well been facing the end of his football career. And he loves football. And he wants that career to continue. So I was impressed with his motivation."
Pace and McCaskey also placed a phone call to McDonald's parents to discuss their son before being comfortable with the move.
"He said, 'If you want, you can talk to my college coach, Urban Meyer, and my position coach from college,' " McCaskey said, according to the Chicago Tribune. "And after talking to him I said, 'I think I'd like to talk to your parents.' And you need to be careful putting too much stock (in that). What would you expect a parent to say about their adult child?
"But the thing that impressed me after talking to them was the support system," McCaskey added. "They go to almost all of his games, even the out-of-town games. They're there for him. He came from a strong two-parent upbringing, which sad to say isn't all that common anymore these days. And even discounting what a parent had to say, I came away impressed with the support system that he has."
When contacted by ESPN.com on Tuesday, the NFL said that McDonald's case is "under review" for possible discipline.
San Francisco cited a "pattern of poor decision-making" at the time it "terminated" McDonald.
McDonald also was implicated in a domestic abuse case involving his fiancee last August. It was announced on Nov. 10 that he would not face charges in that case. The 49ers stood by him at that time.
McDonald has sued the woman who has accused him of assaulting her.
"I feel like what I am doing is the right thing because I know that I am not this bad person that people are making me out to be," McDonald told ESPN last week. "I've been fired from my job. I know some teams don't even want to talk to me because of this past accusation. All I am trying to do is clear my name and move on with my life."
In Chicago, McDonald will be reunited with defensive coordinator Vic Fangio, who was ousted by the 49ers in the offseason and later landed with the Bears.
ESPN.com Bears reporter Michael C. Wright contributed to this report.