Kane has consistently said he wants to win trophies but Spurs are without silverware since the 2008 League Cup and have seemingly gone backwards this season after reaching the Champions League Final last term.
Spurs were already out of every cup competition and languishing in eighth, seven points off fourth-place Chelsea, when the season was postponed in mid-March.
Kane, who is nearly fit again after tearing his hamstring tendon on New Year's Day, has a contract until 2024 with his boyhood club and has previously suggested he could remain in north London for his entire career.
But answering supporters' questions in a live Instagram broadcast with former Spurs midfielder and Sky Sports pundit Jamie Redknapp, Kane refused to rule out leaving Spurs if they are no longer making progress.
"Obviously I get asked this question a lot," he said. "It's one of those things – I couldn't say yes and I couldn't say no.
"I love Spurs, I'll always love Spurs but I've always said if I don't feel we're progressing as a team or going in the right direction, I'm not someone to stay there for the sake of it.
"I'm an ambitious player, I want to improve, get better and become one of the top, top players. It all depends on what happens as a team and how we progress as a team.
"So it's not a definite I'm going to stay there forever but it's not a no either."