Mauricio Pochettino’s men have been linked with the 22-year-old for most of the summer but have so far been frustrated in their efforts.
It had been reported that rivals Chelsea were looking to hijack Spurs in their pursuit of Grealish and Pochettino was lining up a bid for Bournemouth midfielder Lewis Cook instead.
But now, according to the London Evening Standard, Spurs are still leading the race to sign the Villa star.
Villa are thought to be likely to reject an opening bid of £20million for the player but an agreement is still expected to be concluded before next week’s transfer deadline.
The Championship outfit were strapped for cash earlier in the summer with Grealish a certainty to be sold in a cut-price deal.
Billionaire investment though in the form of Nassef Sawiris and Wes Edens has since changed things, with the club now saved from falling into administration.
Whilst players no longer need to be sold, the midfielder himself has indicated that he is interested in a move to Spurs and wants to test himself in the Premier League.
The north London club are yet to make any signings so far in this summer’s transfer window and are the only club in the top six yet to do business.
Discussing their transfer plans on Wednesday, Pochettino said: “I don’t know if we are going to sign or not. We’re working on that.
“I cannot tell you if we’re going to sign one, zero, two or three or four.”
Meanwhile, Toby Alderweireld, Mousa Dembele and Danny Rose could still leave before the deadline next Thursday, while Fernando Llorente and Vincent Janssen are also for sale.
Spurs kick off their Premier League season with a trip to Newcastle on August 11.