Chelsea are on the cusp of appointing Pochettino as their new manager and a verbal agreement is already in place between the club and the Argentinian.
He seems to have emerged as the preferred choice and he is now expected to be in charge of the Blues next season.
Pochettino remains a cult figure at Tottenham and his name has been chanted in recent games by Spurs fans and that connection could be seen as an issue by some Chelsea fans who have a massive rivalry with the north London club.
Cole believes the Argentinian fits what Chelsea need at the moment and is expecting the fans to welcome him once the appointment is finalised.
However, he feels his Spurs connection will come to the fore if Chelsea go on a bad run of form at some point.
The former Blue said on BT Sport: “Pochettino fits for what he does and what he can achieve at the club.
“If he comes in, the fans will back him but in turbulent that [Tottenham connection] becomes an issue.
“But what I would say is I do think he is a fantastic manager and I really think if you take the Tottenham situation out of it, he fits for Chelsea perfectly at the moment.”
Interim boss Frank Lampard has failed to get a tune out of Chelsea and has lost his first five games.