The deal is pending a physical, the source said.
Gibson, a 32-year-old right-hander, went 13-7 with 160 strikeouts for the Minnesota Twins in 2019, but he faltered down the stretch as he dealt with the effects of ulcerative colitis. Gibson's ERA ballooned to 4.73 after giving up 26 earned runs in his final six starts before he was relegated to the bullpen on Sept. 15.
His struggles continued as a reliever in September, and he gave up three runs in one inning in his only postseason appearance -- a 10-4 loss to the New York Yankees in Game 1 of the American League Division Series.
Gibson, who relies on his slider for his out pitch, has provided quality innings for the Twins, making at least 25 starts in each of his last six seasons. In 2019, he set career highs for strikeout rate (9.0/9 IP) and average velocity on his fastball (93.8 mph), but Gibson, who has had strong ground ball rates over his career, also had a career-worst home run/fly ball rate.
More than half of the 23 home runs that he gave up last season -- and nearly 70% of the batted balls he allowed with 100-plus mph exit velocity -- came on fastballs, which he threw only about half the time.
Gibson, who earned $8.125 million last season, had a breakout season in 2018, posting a 3.62 ERA with 179 strikeouts in 196⅔ innings over 32 starts -- all career bests. For his career, he is 67-68 with a 4.52 ERA.
ESPN's Keith Law contributed to this report.