Parker has just guided Burnley to promotion from the Championship - at first attempt after replacing Vincent Kompany last summer.
But he could yet leave Turf Moor for Tottenham, says The Sun.
Despite difficult spells with Fulham and Bournemouth, Parker has admirers inside the Tottenham board room, including chairman Daniel Levy.
Levy appreciates Parker's style of play and believes his arrival would bring little disruption in terms of replacing Postecoglou.
A Spurs source said: "It’s going to be a big summer at Spurs. If they can win the Europa League, then it’s very unlikely that Ange will go anywhere.
“He’ll have earned the opportunity to start the new season and to try and turn around Spurs’ fortunes in the Premier League.
“But if he gets off to a bad start then the noise will become unbearable and he could be out by Christmas."
Contingency plans
The insider also said: “There’s obviously a lot of talk and work going on behind the scenes at the club with contingency plans being put in place as to what life would be like without Ange.
“And Scott Parker is a name who is being talked about more and more. The job he’s done at Burnley has underlined, again, what an excellent young coach he is.
“There’s a view among many at the club that he could be a potential boss for Spurs, but the board know if they do replace the current manager they need to get the next appointment spot on.
“It might be the case that it’s a bit early for Parker. But he’s definitely someone who might potentially end up in the Spurs dugout one day.
“Spurs came very close to appointing Parker in 2021 when he was Fulham manager. But, in the end, he just missed out.
"Many years have now passed, and he’s become more experienced, which has made him an even more attractive proposition.”