Sullivan sparked a row with Zola by describing the team's performance in the recent home defeat by Wolves as 'shambolic', but Cole feels the duo are now pulling in the same direction.
He said: "Everyone at West Ham wants what is best for West Ham. I can't see the manager and the chairman wanting different things. We all want the same thing.
"It is so tight down there at the moment. One game can make a whole world of difference. We are working hard to get that first win. We have not had a good run and we want to work hard and change that.
"Everton is a game where we can get something. I am guy who likes to win his games and it has not been good so far, but we are looking to change that this Sunday."