The skilful Frenchman has long been considered as one of his nation’s top talents, but L’OM’s below par displays last term contributed to him missing out on a spot in Les Bleus’ World Cup squad.
He has since been linked with switches to both Everton and Newcastle, but now it is understood that the Swans have edged ahead of their rivals by contacting the Ligue 1 side to discuss a deal.
Garry Monk has already added Bafetimbi Gomis to his ranks, but with Michu edging towards a move away from the Liberty Stadium he is keen to bring in a creative talent.
Payet, 27, is just that and was one of the bright sparks for Marseille last term, weighing in with eight goals and six assists.
He is valued at around £6.5m by the Mediterranean outfit, who are aware that they may need to sell with financial problems effecting many French clubs.
Everton see Payet as a replacement for Gerard Deulofeu, while Newcastle are known to have good connections on the other side of the Channel.