Payet, who left Marseille in 2015 to join the Premier league club, stood out for his country at Euro 2016.
The Frenchman made it clear he wished to return to Marseille in the current transfer window and British media reported he had refused to play against Crystal Palace two weeks ago.
Although West Ham originally said Payet must stay with the east London club, they agreed to a transfer after accepting a fee of almost 2-1/2 times what they paid for him, though they were still disappointed with the player's decision.
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"The club would like to place on record its sincere disappointment that Dimitri Payet did not show the same commitment and respect to West Ham United that the club and fans showed him, particularly when it rewarded him with a lucrative new five-and-half-year deal only last year," joint chairman David Sullivan said on their website.
"I would like to make it clear that we have no financial need to sell our best players and that the decision to allow Payet to leave was in accordance with the wishes of the manager and the interests of squad unity.
To be frank, my board and I would have preferred for him to have stayed in order to make an example of him, as no player is bigger than the club."
OM, who also recruited French full back Patrice Evra from Juventus in the transfer window, said on their website (www.om.net) that Payet would be officially unveiled on Monday.
Payet, 29, has scored eight goals in 32 games for France, featuring in the Euro 2016 final which they lost to Portugal.
(Reporting by Julien Pretot in Paris and Steve Tongue in London; Editing by Toby Davis and Ken Ferris)
