The striker was scheduled to have coronavirus testing on Monday before starting training on Tuesday.
Kane has had a three-week break after helping England reach the Euro 2020 final.
Spurs have offered no explanation for why the 28-year-old did not turn up.
It is understood Kane felt he had a gentleman’s agreement with Tottenham chairman Daniel Levy that he could leave the club this summer.
A pre-Euro 2020 move did not materialise as he hoped, but Manchester City remain keen on signing the England captain.
Kane’s international’s team-mate Jack Grealish is also a target for City, who have a made a £100m bid for the Aston Villa midfielder, and he turned up for pre-season training with the Midlands club on Monday.
Kane’s contract with Spurs runs until 2024 after he signed a six-year deal in 2018.
He scored 23 top-flight goals last season to claim the Premier League golden boot for a third time, having previously won the award in 2015-16 and 2016-17.
He also has 221 Tottenham goals to his name, which puts him second in their all-time goalscorers’ list behind Jimmy Greaves, who scored 266 times for the club between 1961 and 1970.
Spurs play Chelsea and Arsenal in high-profile friendlies this week and new manager Nuno Espirito Santo has already spoken optimistically about the prospect of working with Kane.
Speaking at his first news conference after being appointed as manager, Nuno said last month: “Harry is our player. Period. No need to talk about anything else, I am looking forward to him joining the group.
“Harry is one of the best players in the world. That is all I need to say.”