May played alongside new boss Ole Gunnar Solskjaer during his time at Manchester United.
And he now thinks his old team-mate should try and force through a move for the France international.
Solskjaer admitted in his first press conference that he would have a say in transfers in January.
“Jose tried to buy Alderweireld and Harry Maguire, they know they need a centre-back. I thought £70-80m on Maguire, who had a good World Cup, was mad,” May told Betstars.
“I think the main issue is that we’re lacking a leader at centre-back. Victor Lindelof could certainly be the number one centre-half, but you need an experience leader. I’m not sure if Chris Smalling has enough presence to play out from the back and start attacks.
“I’d sign Raphael Varane at any cost, £90m to 100m, it doesn’t matter. Y
“You see what has happened at Liverpool with Virgil van Dijk – you know that’s a weak position, go out and improve it at all costs. He’s the one player I think would transform that club.
“If United want to get back to being one of the European leaders, they need to show intent and Varane is a signing that would do that.”
Mourinho was sacked by United on Tuesday with one of his persistent complaints being that he wasn’t allowed a tight reign on transfers.
However, caretaker boss Solskjaer thinks it will be different for him.
Solskjaer said: “I will have an input, of course. The job now is to get to know the players, observe them and see the qualities we’ve got.
“I’ve seen more or less every game from Norway, when I’ve had time from the other job. But to get to know the players and think ‘what can I improve them on’ – that’s my job.
“The club have got their recruitment and scouting, so I’m sure they’ve got their targets. But I’ve not sat down and talked about that, because my job is to get these players enjoying football.”