Zola clashed with Sullivan after he publicly accused the team of being "pathetic" and "shambolic" following their defeat to Wolves.
After West Ham secured their Barclays Premier League survival with a 3-2 victory over Wigan last weekend, Gold said he would like to see Zola continue as manager.
The pair have a cordial relationship, though seemingly not good enough for Zola to be told that Gold was bidding for a new midfielder.
Asked whether he or any manager could be expected to succeed in those circumstances, Zola said: "At the end of the season when I talk to them they will let me know what their regime will be.
"David Gold has always been supportive and he has stayed close to us. It is good. He is always here before the matches and that is something I appreciate."
Asked if Gold had spoken to him about the future, Zola added: "No, we spoke about other things.
"Regarding my future, it is something that will have to be discussed at the end of the season when I will have a meeting with the owners."