Zola's future at Upton Park has been the subject of constant speculation after a disappointing campaign.
A number of names have been linked with replacing Zola in the Hammers hotseat, but Spector feels the Italian has the backing of all the players in the West Ham squad.
"We as players don't know what is going to happen with the manager," Spector told skysports.com.
"But it is clear that all the players like him as a person and as a manager.
"We are behind him and I think our celebrations against Wigan showed our solidarity with the manager.
"The manager believed in us and we believed in him."
Spector admits West Ham have underachieved this season and that their survival came as a huge relief after a tense campaign.
"Staying up was a massive relief as it has been a stressful season," added Spector.
"There was a lot of uncertainty around the club and it seems West Ham were getting more bad publicity than other teams so the team did well to stick together.
"It has been a poor season, as definitely we could have done better.
"We have a number of great players and we feel we did not hit the heights this season."