ESPN reported last week that the two clubs were in talks over the proposed move.
Morris, 26, left U.S. men's national team training earlier week ahead of the move and will join Swansea for the remainder of the Welsh club's Championship campaign.
"At the end of the day, this move is all about Jordan and what he wants from his career," Sounders general manager & president Garth Lagerwey said in a release. "Our club and city are behind him as he embarks on the next phase of his journey, and should he return to Seattle at the end of this loan, we know he will be stronger for the experience."
Swansea are currently second in the Championship on 46 points through 24 matches as they look to secure a return to the Premier League for the first time since 2017-18.
Morris, who grew up in near Seattle in Mercer Island, Washington, has spent his entire professional career with the Sounders, spurning offers from Werder Bremen when he signed with the MLS club in 2016. He has scored 42 goals in 129 league, playoff and cup appearances. During his tenure in Seattle, the Sounders claimed two MLS Cups and reached the final on two other occasions, including the 2020 season when Seattle fell 3-0 to the Columbus Crew.
"I'm very proud of the player and professional that Jordan has become. He's been a large part of this club's success and has more than earned this opportunity to ply his trade abroad," Sounders head coach Brian Schmetzer said in a release. "This particular move gives him a chance to showcase his skillset and opens a number of options for his future. On top of that, I am excited to watch him play for Swansea."
At international level, Morris has been impactful as well, scoring 17 goals in 39 appearances. He was part of the side that won the 2017 Gold Cup, scoring the game-winner against Jamaica.
Swansea's next league match is against third-place Brentford on Wednesday.