As part of the trade announced Tuesday, the Blue Jays agreed to send cash to Chicago to offset some of Phelps' contract, which includes a $2.5 million base salary for this year, a $1 million club option for 2020 and numerous performance bonuses.
Phelps will cost Chicago a maximum of $1.25 million this year. The Cubs are responsible for the $819,892 remaining of his 2019 salary and the first $430,108 of the performance bonuses Phelps earns.
"Very aggressive sort," Cubs manager Joe Maddon said. "Very physical, aggressive, let's-go kind of guy. It's going to be wonderful to work with him tomorrow."
The 32-year-old tore the ulnar collateral ligament in his pitching elbow on March 17, 2018, during his final pitch in a spring training outing against the Los Angeles Angels, had surgery March 26, 2018, and did not return to the major leagues until this June 17. He had a 3.63 ERA in 17 relief appearances with the Blue Jays, going 0-for-2 in save chances.
He is 30-33 with a 3.88 ERA over 245 games in seven seasons with the New York Yankees (2012-14), Miami (2015-17), Seattle (2017) and Toronto.
Cubs right-hander Steve Cishek was teammates with Phelps in Miami and Seattle.
"He's good. He's done it for a long time," Cishek said. "Good locker-room guy. He's been through some fire before."
To open a 40-man roster spot for Phelps, the Cubs moved left-hander Xavier Cedeno to the 60-day injured list.
The Cubs also placed right-handed reliever Pedro Strop on the 10-day injured list with left neck tightness and called up right-hander Duane Underwood from Triple-A Iowa.
"Sunday, I was playing catch and my neck was feeling weird," Strop said. "It was really tight. I woke up today, and it wasn't feeling that good, either."
Phelps could earn $2.5 million in bonuses this year based on games: $250,000 each for 25, 30 and 35 games, and $350,000 apiece for 40, 45, 50, 55 and 70. He also could earn $750,000 this year based on games finished: $125,000 each for 25 and 30, and $250,000 apiece for 35 and 40.
His option price would increase to $3 million if he pitches in 30 to 39 games this year, to $5 million if 40 to 49 games and to $7 million if 50 games. He has differing bonus chances for 2020 depending on his base salary that year.
Hatch, 24, was 4-10 with a 4.59 ERA this season at Double-A Tennessee. Selected by Chicago in the third round of the 2016 amateur draft with the 104th pick, Hatch signed for a $573,900 bonus. He is 17-27 with a 4.10 ERA in 73 starts over three minor league seasons.
ESPN's Jesse Rogers and The Associated Press contributed to this report.