The England international, a £35million signing from Newcastle two years ago, made the move from Anfield to Upton Park on a season-long loan deal in the summer.
Sullivan told the Daily Mail: "I am often asked about Andy Carroll's deal. It is a season-long loan. Liverpool had an option to take him back in January, but he has been injured.
"West Ham have an agreement with Liverpool to sign him at the end of the season, but we have no agreement with the player.
"It was his desire to come to West Ham to prove himself to Liverpool and that hasn't changed.
"He has had injuries since he came, but he is back to fitness now and we have 15 games left... we need some goals and some points, so let's see."