A source told ESPN The Magazine's Buster Olney that Morse will receive $5 million plus incentives for the 2014 season.
The deal is pending a physical, assistant general manager Bobby Evans said Thursday. Traded from Seattle to Baltimore on Aug. 30, Morse also can play first base and right field, and he gives manager Bruce Bochy some flexibility in the lineup.
Morse, 31, missed time this past season with a broken right pinkie and strained right quadriceps before returning from the disabled list in late July. The nine-year veteran batted .215 with 13 home runs and 27 RBIs in 88 games with the Mariners and Orioles.
He will fill a void at a position the Giants had been determined to upgrade this winter.
Morse will join an outfield featuring Angel Pagan in center and Hunter Pence in right. Pagan received a $40 million, four-year deal last December, and Pence was rewarded with a $90 million, five-year contract before this past season ended. He played in every game in 2013.
Information from The Associated Press contributed to this report.