West Ham are said to have offered £36 million and Javier Hernandez for the French international.
If he did join it would be an intriguing signing, and one which Dimitri Payet may not be best pleased with.
Thauvin and former West Ham man Payet have linked up to good effect at Marseille, with the duo two of the French side’s most creative players.
And Thauvin has previously admitted that he didn’t even think he could get on well with Payet at first.
"I thought that with Dimitri (Payet) we didn't like each other and, in the end, we like each other. If I had let my pride speak, I would not be where I am today. In the end, it was good for both of us,” he said to MaxiFoot.
If Thauvin did move to West Ham, the London Stadium crowd may hope that he has a similar impact to what Payet did there.
Payet’s exit from West Ham was acrimonious, but he remains one of the most exciting players the Hammers have had in recent memory.
At the moment the Thauvin rumours have to be questioned though.
It seems unlikely that West Ham would be offering up a striker when they are so short, to take another winger, when Manuel Pellegrini’s side are well stocked out wide.
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