The Premier League has been suspended until at least the end of April due to the ongoing coronavirus pandemic.
There are fears even the set period for resumption might have to be extended further and Premier League games might have to be played behind closed doors if there are any hopes of completing the season.
Kane believes further extensions will be detrimental to the footballing calendar, which has already seen major international competitions postponed to make way for domestic games to be fulfilled, and there must be a cut-off point where the season must be cancelled.
"I know the Premier League are going to be trying all they can to finish the season, looking at every option possible. For me, I feel we need to try to finish the season, but there needs to be a point where enough is enough," he said on a live Instagram broadcast with Sky Sports pundit Jamie Redknapp.
"We may have to say, 'we can't finish it'. Going into July and August and pushing next season back, I don't really see too much benefit in doing that. I don't know what's going on behind the scenes, and what it means financially for clubs.
"From a players' point of view, let's try to finish the season, but there needs to be a limit and probably the limit for me is by the end of June.
"If it isn't completed by the end of June, we should look at the options and look forward to next season otherwise I feel it will have such a knock-on effect to next year, the Euros, then the World Cup in 2022."
Kane’s Spurs side are eighth on the Premier League log.
