Terry worked with Conte at the Blues, where they won the Premier League and FA Cup together during the 2016/17 season - which was the centre-backs last at Stamford Bridge.
The Italian is known to be a tough taskmaster who is famed for his gruelling training regime to get his players into shape.
Conte was hired by Spurs this week as the club move swiftly to replace Nuno Espirito Santo.
Terry wrote on Instagram: "I hate to say it but what a signing this is. The players will love Antonio and his staff. Be ready to go to work @HarryKane."
Conte will be tasked with turning things around at Tottenham after disappointing performances and results under his predecessors.
Spurs currently sit ninth in the Premier League with five wins and five defeats from their opening 10 games.
Former Bolton, Everton, Newcastle and West Ham boss Sam Allardyce believes the 52-year-old will whip the players into shape.
He told talkSPORT: "If anyone is going to get Tottenham into shape, that man will. I've known him for quite a while and this man takes no messing.
"His views on the game and way he portrays it and goes about it brings results.
"They brought Jose (Mourinho) in for that but it didn't work and this time they have to go his way and I think he would have demanded that.
"Tottenham will improve, whether they will improve enough to be a top four side now is the big question.
"They will definitely improve under Antonio Conte and he is very good at what he does, very disciplined.
"A little old school and disciplined in how the players train and how they handle themselves, but his track record speaks for itself."