The 21-year-old has been a target for both Liverpool and Tottenham, both of whom are looking for a left-back.
Davies played no part in Swansea’s 2-0 defeat against Minnesota United on Saturday and it is understood that he has passed his medical and travelled with the Spurs squad to Toronto.
A fee reportedly in the region of £10m was thought to have been agreed between the two sides, while Tottenham midfielder Gylfi Sigurdsson is on course to move in the other direction.
The Iceland international has held positive talks with the club where he spent half a season on loan in 2012.
Sigurdsson joined Swansea from Hoffenheim in January 2012 and quickly became a fans’ favourite, scoring seven goals and providing four assists in 18 Premier League games.
Swansea thought they had agreed a deal to sign the midfielder at the end of that season but Brendan Rodgers’ departure to Liverpool scuppered their hopes of signing the player on a permanent basis.
Despite Rodgers’s desire to sign Sigurdsson for Liverpool, the 24-year-old opted for a £8.8m move to White Hart Lane in July 2012.
The former Reading midfielder has made 83 appearances for Tottenham in all competitions, scoring 13 goals and contributing nine assists.