Carroll is yet to feature for the Hammers this season after successive ankle injuries that has seen him miss over seven months and the former Liverpool man is not expected back until the new year.
After a successful loan spell at Upton Park last season Carroll joined the struggling East Enders in the summer for in excess of £15million, a club record fee.
With a real lack of cover in the squad, West Ham have failed to repeat their fine form of last season and are well and truly in a relegation scrap this time around.
With January on the horizon, Sullivan knows that the club have to find another striker to bring in but financially Carroll has ruined them this season.
“Had we known he would be out for this long, we would not have signed him,” Sullivan told the BBC’s Football Focus.
“We are not a rich enough club to deal with that. You know any player can get injured, but we can’t buy a player knowing he is going to be out for half the season.
“When we signed him we were assured by the medical staff that the very, very latest he would be back was 1 September. That would have meant he would only miss two league games.”