Former England international Sheringham spent five goal-laden years at White Hart Lane before jumping at the chance to join the Red Devils in 1997.
Four seasons at Old Trafford delivered seven pieces of silverware, with three of those secured during a memorable 1999 Treble-winning campaign.
Some 20 years on and the man currently leading the line for Spurs is being heavily linked with a similar switch to benefit his career.
Sheringham, though, believes that Kane can fulfil his ambitions in north London – as can team-mate Dele Alli – with there no need to go looking for success elsewhere.
He told The Sun : "Tottenham wasn’t a stepping stone for me. I wanted to get to Tottenham and have what Harry Kane has got now.
"This has to be the most exciting time at Spurs since Terry Venables. There’s no need for Harry to go anywhere else.
"The way the club is going, it could be the biggest in London.
"Why would he want to go to Manchester, Barcelona or Real Madrid, when everything is here for him?
"Dele Alli, exactly the same - performing at the top of his game, a regular for England.
"They don’t need to leave to win trophies. You don’t have to go anywhere else when Tottenham are progressing."
Kane is currently the Premier League’s top scorer with 19 goals to his name.
He has 24 in all competitions and will be looking to beat his previous personal best of 31 before the 2016-17 campaign is out – with those efforts helping Spurs to remain in contention for a top-four finish and a return to the Champions League.