Spurs have reportedly submitted an offer of £27 million for the attacking midfielder, who is also interesting Napoli.
Suarez, 23, was formerly part of Manchester City's academy, and Tottenham's interest should cause Pep Guardiola to consider also entering with a raid on his former club.
Suarez left City in 2013 after two years in their academy, having made two first team appearances.
The youngster spent last season on loan at Villarreal, scoring three goals and providing seven assists.
Just as they were when he left, City are spoiled for choice in attack, but Suarez is a better and more mature player.
He told The Observer last year: "You can see that Pep’s giving young players a chance. When I was there it was a bit different."
So with title rivals Tottenham eyeing up the talented 23-year-old, City might make a mistake in simply allowing Mauricio Pochettino's side to move in and snap him up.
Having already had to pay Spurs £50 million to sign Kyle Walker and watch them replace him with former City academy defender Kieran Trippier, it will not be a good look on City's part if Suarez goes to Spurs and becomes a big star.