The England captain was due back at the club’s Hotspur Way training centre for a Covid-19 test following his three-week break after the European Championship.
However, the PA news agency understands Kane did not arrive as planned, throwing his future further up into the air.
Spurs – who face Manchester City at home on the opening day of the season – have declined to comment.
The 28-year-old is keen to leave the club this summer in order to win trophies, but with three years left on his contract, Tottenham have no intention of selling their prized asset.
Kane hinted at the end of last term that he would like to join Manchester City, but the Premier League champions have previously indicated they would not pay the sort of £150million fee required to even tempt Spurs to the negotiating table.
Their pursuit of Jack Grealish for £100million is also reported to impact their ability to sign Kane.
Kane has been in the Bahamas following England’s run to the Euro 2020 final and looks to have set his stall out to force a move from the Tottenham Hotspur Stadium.
New boss Nuno Espirito Santo, who was assured Kane would not be sold during discussions for the job, said last month that Kane could be “counted on” next season.