The Telegraph was the first to report Swansea's interest in Morris.
Morris, 26, is taking part in a training camp with the U.S. men's national team. Earlier this week he addressed his future, saying there "are definitely some conversations" between Seattle and European clubs.
"I think that for me, it's a similar stance of kind of where I was before MLS Cup. If the right offer and the right team comes forward, definitely it's something that I'd be interested in," Morris said earlier this week. "And if things don't work out, and then I end up in Seattle, I love it there and I feel like I'm growing a lot in Seattle."
Columbus Crew dominate Seattle to win MLS Cup 2020
Columbus beat reigning champions Seattle Sounders 3-0 to win their second MLS Cup in club history.
Morris, who grew up in nearby Mercer Island, Wash., has spent his entire professional career with the Sounders, spurning offers from Werder Bremen when he signed with the Sounders in 2015. 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.
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 are currently in second place in the English Championship, four points behind leaders Norwich City.