When Kane was injured twice with the same problem last season, he missed seven weeks from September, then just over three weeks from March.
With the World Cup in Russian Federation now less than 100 days away, the FA will not want Kane rushed back from his injury by Spurs.
Southgate is set to name his next England squad for the friendlies against Holland and Italy at 2pm on Thursday.
It means for club he is likely to miss at least four Premier League games, the FA Cup quarter-final with Swansea and a semi-final should they make it that far.
While the England striker is also set to miss Spurs' key Premier League fixtures against Chelsea and Manchester City next month.
However, other reports said it has not yet been possible to discover how long Kane will be out because he hasn't had a complete scan due to swelling around the joint.
Both Southgate and Spurs boss Mauricio Pochettino will be praying Kane's recovery is more in line with when he hurt his ankle for the second time, against Millwall past year, but recovered much faster than expected.
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Honda X-Blade 160 launched at Rs 78500
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