The deal was first reported by The Washington Post.
Morse hit .291 with 18 home runs and 62 RBIs for the Nationals in an injury-plagued season, playing mostly as an outfielder.
"With the addition of Mike Morse, along with the other veteran players we have added, we believe this will help us be a more complete offensive team," Mariners' general manager Jack Zduriencik said in a statement announcing the trade. "We wanted to add additional right-handed power and Mike is a major step in that direction."
Jaso, 29, batted .276 with 10 homers and 50 RBIs but became expendable when Seattle signed Raul Ibanez to be its designated hitter, a move that shifts Jesus Montero from a catcher-DH split to primarily behind the plate. The AL West champion A's had Kurt Suzuki as their main catcher last season before trading him to the Nationals.
Cole, a 21-year-old right-hander, was among four players traded by Washington to Oakland in December 2011 for pitcher Gio Gonzalez. Cole was 0-7 with a 7.82 ERA in eight games at Stockton of the California League and 6-3 with a 2.07 ERA at Burlington of the Midwest League, both Class A level.
To open a roster spot, Oakland designated catcher George Kottaras for assignment.