The Welsh giants are keen to add to their attacking force during the ongoing transfer window and have identified the 26-year-old as an ideal candidate.
Steve Cooper’s side are currently locked in talks with Major League Soccer outfit Seattle Sounders to sign Morris on a six-month loan deal.
The United States international previously attracted offers from Bundesliga sides, but looks set to join Swansea in their fight for promotion.
The signing of Morris would be viewed as a major coup for the Championship side as there were European clubs interested in the player after he indicated he could leave Sounders in December.
However, Morris is currently with the United States national team at a training camp in Florida and could only arrive at the Liberty Stadium next week.
The 26-year-old scored 12 goals and provided nine assists from 29 appearances across all competitions for Sounders last season.
Swansea currently sit second in the Championship table and Cooper will be hoping that Morris’ addition boosts their chances of promotion.
