The Sun says new Hammers boss Sam Allardyce is fighting a losing battle to hold on to the Senegalese striker, despite offering him a £15,000-a-week pay rise.
Ba, 26, has a get-out clause because of relegation, which allows him to escape the remaining three years of his current £35,000-a- week contract.
Sunderland have also been chasing the club's most sought-after player and it is causing bad blood.
West Ham are furious at the pursuit of their star and have already reported the Wearsiders to the Premier League over claims of an illegal approach.
Everton finished seventh last season but boss David Moyes is keen to add more firepower to his team.