Kane damaged his ankle ligaments for the third time in 18 months against Bournemouth at the weekend and was seen leaving the ground on crutches.
Kane got his foot caught under goalkeeper Asmir Begovic after half hour at Bournemouth and had to be taken off. Mauricio Pochettino afterwards admitted it left him anxious.
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.
He was injured when his right foot got trapped under Cherries keeper Asmir Begovic.
Mauricio Pochettino's side travel to Swansea for an FA Cup quarter-final on Saturday before...
Kane will also have to sit out England's two friendlies this month against Holland and Italy, with manager Gareth Southgate naming his squad on Thursday.
Gareth Southgate is set to name his squad for this month's worldwide friendly matches against Holland and Italy, but Kane will be absent for both ahead of this summer's showpiece.