It is understands that a proposed fee, believed to be around €23m, was rebuffed by co-chairman David Sullivan yesterday afternoon.
Payet shocked the Hammers last week by declaring his desire to leave the London Stadium, despite having only signed a fresh deal last February after shining in his first season in the Premier League.
Despite his stance, both co-chairmen Sullivan and David Gold insist they have no financial need to sell the former Lille man and are refusing to be bullied into a quick sale.
The first team were given yesterday off following their impressive display against Crystal Palace on Saturday. It is believed manager Slaven Bilic has followed through on his threat to banish Payet from training - and the Frenchman will now practise with the under-23 side instead.
With other top-flight clubs also seeing their leading players linked with moves away in the current transfer window, Gold suggested how he would deal with the situation.
“With a number of top Premier League players holding clubs to ransom is it time to close the January transfer window?” Gold wrote on Twitter.
Elsewhere, Robbie Keane could join Preston North End’s growing Irish contingent, with the striker being linked with the Championship club.
West Brom have made a £10m offer for Hull midfielder Jake Livermore. The Baggies are chasing the 27-year-old after missing out on Morgan Schneiderlin to Everton.
They made an offer of around £18m for Schneiderlin but he joined Everton from Manchester United in a deal worth £24m.
There was also interest in Jeff Schlupp, but the Ghana international moved to Crystal Palace from Leicester for around £12m last week.
Chelsea manager Antonio Conte says he will check on the fitness of Diego Costa, having refused to confirm he is attempting to force a move to China.
Conte also said if there were any issues with players, he would resolve them behind closed doors.
“If there is a problem - and I repeat ‘if’ - with the players then I prefer to sort the problem in the changing room, not outside and not in press conference.”
Meanwhile, the Chinese Football Association has announced reforms which will see only three non-national players per game allowed to be fielded by Super League clubs.
In a move which could impact on further stellar signings, the Chinese FA confirmed that new rules would be brought in from the start of the 2017 domestic campaign, which is scheduled for March.
The changes will also see two Chinese players aged under 23 named in matchday squads, at least one of whom must start the game.
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