Kane turned his ankle in a challenge with Asmir Begovic.
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The 24-year-old hurt himself during the 4-1 win over Bournemouth on Sunday and was supposed to be sent for a scan on Monday but swelling has continued to cause a delay.
On the first occasion he suffered the same injury he missed seven weeks - a similar period to the expected lay-off this time - in September 2016, while the second injury only ruled him out for three weeks.
Kane, who has scored 35 times for Tottenham this season, would only have a handful of matches remaining this term to prove his fitness to England manager Gareth Southgate.
Tottenham return to action against Swansea in the FA Cup at the weekend without their talisman.
As it stands, Kane would miss Spurs' FA Cup quarter-final at Swansea on Saturday, a semi-final if they progress and four league fixtures after the worldwide break - a potential top four decider against Chelsea and further clashes with Stoke, Manchester City and Brighton.
He will then miss the semi-finals if they win and four league fixtures after the worldwide break, a potential top four decider against Chelsea and further clashes with Stoke, Manchester City and Brighton.