The 23-year-old England international moved to Upton Park on a season-long loan deal last week.
Liverpool paid £35 million for Carroll in January 2011 but the former Newcastle forward struggled for form at Anfield. He has now moved to West Ham and impressed on his debut against Fulham on Saturday before suffering a hamstring injury.
Liverpool boss Brendan Rodgers had hoped to sign a new forward before the transfer window closed but no deals materialised, sparking suggestions he could be forced to recall Carroll.
It is understood Liverpool can cancel the loan agreement in January but West Ham co-owner David Gold took to Twitter to say he wants Carroll to stay for the entire campaign.
His Tweet read: "The exact terms of Carroll's year-long loan deal are subject to a confidentiality agreement but we are hopeful he will be with us the whole season."
With Carroll gone until at least the new year and no replacement signed before the transfer deadline passed, Rodgers has been linked with a host of free agents to fill the void left by Carroll's departure.
However, a move for former Anfield hero Michael Owen now appears to be a non-starter.