Postecoglou has steered the Parkhead outfit to their 53rd Scottish top-flight title this season and the Bhoys are now one step away from clinching a record eighth domestic treble.
Spurs, on the other hand, witnessed lacklustre performances this term and they missed out on qualifying for next season’s European football.
The north London outfit are now active in the market to search for a permanent manager for next term, and they are heavily linked with Postecoglou as their next manager, following their defeat in striking a deal for Feyenoord boss Arne Slot.
And Sutton believes that the Celtic manager could be very eager to go to Spurs this summer as he did not extinguish the prospect of becoming Tottenham manager when he had the chance to do so.
“I don’t know what is going on in Postecoglou’s head”, Sutton wrote in his Daily Record column.
“I’m only giving an opinion.
“His cards seem to be always close to his chest.
“For all I know, he’s perfectly content to stay at Celtic.
“And of course, Postecoglou could have put these Spurs links to bed last week.
“Whether it’s as simple as he didn’t do it so therefore he must be mad keen to go there, I can’t say.“
It is suggested that Postecoglou is on a 12-month rolling contract with Celtic amid the reverberating demand to hand him a long-term contract.