Gill, speaking on BBC Radio's Five Live programme, said they were discussing paying a fee of around 25 million pounds for the player.
"The actual situation will be cleared up in early June and without a doubt we will look at it," Gill added.
"We will assess the squad. He has done very well, come on and made an impact in some crucial games this year and we would like him to stay, but the actual economic situation needs to be clarified."
Asked if he and manager Alex Ferguson wanted Tevez to stay at Old Trafford, Gill replied; "If it can be structured correctly, yes we do."
He also said there were no ongoing discussions about Cristiano Ronaldo leaving the club, who won a record-equalling 18th English League title on Saturday.
COMPLICATED DEAL
Gill said the deal for Tevez was more complicated than usual because his "economic rights" were owned by the agent Kia Joorabchian.
Gill explained that United had paid a "loan fee" to Joorabchian for the last two years but have to decide now whether to pay a fixed fee at which point Manchester United would own Tevez and have a new contract with him.
However, if United do not agree a deal, Tevez could be offered to other clubs by the company which owns him.
Tevez appeared to be waving goodbye to the Old Trafford faithful when he was substituted during Saturday's 0-0 draw with Arsenal.
Asked if he was waving goodbye, an emotional Tevez told Sky Sports: "Maybe goodbye, maybe.
"I like the fans so much. Thank you. It was very difficult for me," he added.
Gill said the 67-year-old Ferguson, manager at Old Trafford since 1986, had no intention of retiring in the near future.
"He is still very motivated to go on and win things with Manchester United," Gill said. "He is very well supported by the staff and players and still dedicated 100 per cent to Manchester United.
"Hopefully, he will go on for many years."
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