Marseille will ask for £6.5 million if they are to allow the 27-year-old to depart the club this summer and Swansea are hoping to wrap up a deal soon.
Everton are also interested as they look to fill the void left by Gerard Deulofeu's return to Barcelona, while Newcastle have been following the French international for some time too.
Belgian winger Roland Lamah has returned to Spain with Osasuna following a loan spell with Swansea, and Gary Monk is keen to bring in Payet ahead of his first full campaign as manager.
The former-Lille man can play on either side, and has made seven appearances for the French national team but couldn't make the final 23-man squad for this summer's World Cup and could be set to leave French football for the first time in his career.
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He made 36 appearances for Marseille last season, scoring eight times with six assists and would add creativity to a Swansea side keen to improve on last season's 12th placed finish.
Payet only joined Marseille last season and made an immediate impact, scoring a brace within 15 minutes on his debut, but the French club are willing to let him leave if they receive the right offer and Swansea are leading the chase.