The Reds had a £20m offer for Bowen rejected last summer.
Although the Hammers seem reluctant to sell their 24-year-old striker, the Reds are unfazed and will make a second bid for him in the January transfer window, says the Liverpool Echo.
In both West Ham's games against the Red Devils, which include their 2-1 loss in the Premier League, and their 1-0 win in the Carabao Cup, Bowen was one of the stand-out stars.
Although he didn't get on the score-sheet, Bowen set up one goal and had six shots on target over the course of the games.
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