The attacker was being linked to Liverpool in a big money move during the transfer window.
However, the Reds are now pursuing Raphinha from Leeds United.
According to Give Me Sport, West Ham are very pleased at the news, as they want to hold onto Bowen under any circumstances.
Manager David Moyes wants to offer him a new contract this year, which will raise his wage and keep him at the club for many years to come.
Bowen's present deal runs out in 2025.