United boss Jose Mourinho is said to be a fan of the French playmaker and keen to sign a wide-player when the transfer window reopens in January.
The Sunday Mirror claim West Ham would want at least £35m and resist any bids in January but might lose Payet in the summer.
The newspaper also say Sunderland defender Lamine Kone could be on his way to the London Stadium.
The Hammers, it is claimed, are lining up a move for the Ivory Coast international, who will apparently cost around £15m.
Meanwhile, Chelsea want Hoffenheim defender Niklas Sule, again according to the Sunday Mirror. Everton are also said to be interested in the Germany international.
Tottenham boss Mauricio Pochettino has launched a bid to bring Utrecht’s France Under-21 international Sebastien Haller to White Hart Lane, according to the Sunday Express.
It is claimed that striker Haller, 22, is keen on a move and has been made available for around £7m.
The Sunday Express also claim Chelsea will battle with Manchester United to sign Real Madrid’s James Rodriguez.
And The Sun on Sunday say keeper Thibaut Courtois will consider running down the remaining 18 months of his Chelsea contract amid apparent interest from Real Madrid.