The report says Everton’s failure to land a frontman after pocketing £75m selling Lukaku to Manchester United sparked tension between Koeman, director of football Steve Walsh and Goodison chairman Bill Kenwright.
Carroll has 18 months left on his £85,000-a-week deal and West Ham could be prepared to cash in if they get the right offer in January.
Everton are currently losing at half time having won just twice in the Premier League this season and are currently below the Hammers in the league table.