But unless a deal can be reached before he meets up with Southgate’s England squad, he is likely to have to wait until later in the summer to secure what is expected to be a British record transfer in excess of £100million.
Speaking after announcing his provisional European Championships squad, Southgate said he expects Kane’s full focus to be on England.
He said: “I don’t feel I need to talk to Harry Kane regarding focus or professionalism. He has one goal with us and that’s to win the European Championships.
“He knows how important that is at this moment in time. What’s happening at his club is a matter for him and Tottenham.
“He’s made some statements in the last few days. It’s very unlikely that transfer deals will be done while we are away.
“It’s different when we meet up in September and I’m very realistic that you can’t stop phones ringing, but most deals will be done at the end of the tournament or in the first week before it starts.
“I would expect most deals in terms of our players to be done after the tournament.”
Any move for Kane is unlikely to be a simple process with Levy a notoriously fierce negotiator.
He is likely to hold out for as much as £150million for his prize asset, with United, City and Chelsea the only English clubs that could afford the England captain.
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All three clubs are desperate to bring in a world class striker this summer and the three-way battle for his signature could only elongate the process.
Barcelona and Real Madrid are also huge admirers and could yet lure him away from England.