The Tottenham striker was taken off in the first-half of their fixture against Bournemouth on Sunday with damaged ankle ligaments - the third he has suffered the same injury in 18 months.
Based on the initial diagnosis, Kane may be ready to return in time for Spurs' final three scheduled league games - against Watford on the last day of April, West Brom and Leicester.
Winks' game-time at Spurs has been hampered by a persistent ankle injury, which he first suffered at the end of last season.
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But, depending on their FA Cup progress and the dates fixtures get rearranged for, he could have two bonus opportunities to build up his fitness ahead of the World Cup.
He is also set to be absent from England's two friendlies against the Netherlands and Italy this month.
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.
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 a year ago, but recovered much faster than expected.
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England will face Tunisia, Belgium and Panama in the World Cup group stage in Russian Federation.
Spurs are set to travel to Swansea for their FA Cup quarter-final this weekend without Kane and face Chelsea in the Premier League on April 1.
Kane's history of injuries to his right ankle is a concern.
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