The Swans have made no secret of the interest in the Tottenham right-back.
Naughton started the first game of the season against West Ham United but was sent off at Upton Park during their opening day victory at Upton Park.
Summer signing Eric Dier filled in for him during his suspension in the 4-0 win over Queens Park Rangers that followed, and kept his place during yesterday's 3-0 defeat at home to Liverpool.
Naughton has played in both of their Europa League games against AEL Limassol and Spurs still have Kyle Walker to welcome back to the side when he recovers from his lengthy spell on the sidelines with a pelvis injury.
Swansea boss Garry Monk has been open in his interest in Naughton this summer, and they made an approach for the 25-year-old before the Premier League season got underway.
Whether Mauricio Pochettino will be prepared to let his right-back leave with only hours left before the transfer window shuts remains to be seen, but it would take a lot of work if he wants to bring in a replacement.