Spurs have been frustrated so far in their efforts to land the highly-rated 22-year-old.
Villa have rebuffed offers up to £25 million ($32m), with the Championship club reluctant to part with prized assets despite their much-publicised financial struggles.
Redknapp, though, believes that another approach from north London will be made before the summer transfer window slams shut.
He told talkSPORT on the possibility of a deadline day deal being done: “Daniel [Levy] is a shrewd operator, don’t you worry.
“He’ll know what’s going on with the Jack Grealish deal, for sure.
“I think it’ll be a last minute job.”
Spurs are yet to make any additions to their ranks this summer, with Mauricio Pochettino still working with the same squad that secured a third-place finish in 2017-18.
That may be considered a little surprising, but the club has invested considerable funding into the reconstruction of White Hart Lane.
They are also already blessed with a star-studded squad which boasts strength in depth across the board.
Redknapp feels that is the reason for a lack of movement, with it difficult to tempt fresh faces into a move if they know they will not be guaranteed regular starting berths.
The former Tottenham manager added: “They’re not that far short.
“If you look at Tottenham’s squad, they’ve got great cover in most positions.
“The only area they haven’t really got cover for is Harry Kane.
“But if you’re looking to bring a striker in, you’re basically going to tell him you’re not going to play – that’s the problem for Spurs.
“Even Grealish, it’s a gamble for him if he goes to Tottenham. I like him, he’s fantastic talent, but is he going to play?”