Barcelona-based newspaper Mundo Deportivo claim Tottenham are open to selling Kane.
However, Levy wanta the deal done on his terms.
Tottenham are desperate to keep Kane and have always maintained their insistence he will not leave the club.
But it is claimed Levy is concerned Kane could soon down tools in order to force through a transfer.
That is what happened in 2012 when Luka Modric decided he wanted to quit White Hart Lane for Real Madrid.
And Mundo Deportivo claim Levy doesn’t want a repeat of that situation.
According to the report, Spurs are willing to negotiate with Real so long as they play to their rules.
The Premier League side would demand a ‘stratospheric’ fee in excess of Neymar’s £200million transfer record, the report suggests.
Other reports in Spain have claimed Modric could be used as part of a cash-plus-player deal for Kane.
But Tottenham are said to only want a straight transfer – if they are to sell up at all.
Kane has been a revelation at Spurs since breaking into the first team in 2014.
He is one goal off his 100th in the Premier League and already boasts 34 for Spurs and England this season.
Real are expected to sell at least one of their strikers – Karim Benzema, Cristiano Ronaldo or Gareth Bale – this summer.
There were reports in January that Manchester United were weighting up a deal for Kane but Spurs are unlikely to want to negotiate with a Big Six rival.
Mundo Deportivo claim a cash-plus-player deal involving Bale would not suit Spurs as they could not be able to match his £10.5m-a-season wages.
Real president Florentino Perez is said to be on standby in case Spurs do decide to sell their talisman.
Interest from United is cooling but Paris Saint-Germain remain another option for the 24-year-old.
The report adds Eden Hazard remains on Real’s radar, while they maintain an interest in Neymar even if a deal would seem improbable.