The club have reportedly fielded interest from Chelsea and are prepared to listen to offers for the striker before the end of the January transfer window.
Carroll was not included in the squad on Tuesday as West Ham edged into the FA Cup fourth round after a 1-0 replay win over League One side Shrewsbury.
Shrewsbury held their Premier League opponents to a goalless draw in the opening match last week and thwarted them for another 90 minutes at the London Stadium before Reece Burke struck in extra time.