The newly-promoted side were close to signing Carroll from Liverpool in a loan deal earlier this month but Sullivan admits that a transfer would have been to expensive for the Hammers.
However, Sullivan and David Gold were prepared to give the move the green light before Carroll decided to fight for his place at Anfield.
Sullivan told The Mirror: "It was a terribly expensive deal with the loan fee and with his wages, which were enormous - too much for a club like ours.
"But, as the manager really wanted him, we were prepared to pay it.
"Then there was a huge transfer fee which would have blown our budget for several years."
West Ham are now believed to be interested in signing Matt Jarvis from Wolverhampton Wanderers.