Kane seemed to have pulled Spurs level after Junior Stanislas opened the scoring on the south coast, but with his goal ruled out for offside, concerns grew after the England striker remained hunched on the turf clutching his ankle.
Spurs boss Mauricio Pochettino said: "I am concerned until he tomorrow or after tomorrow makes the scan. We need to wait until tomorrow".
The Bosnian goalkeeper's miserable afternoon was compounded in injury time when he could only swat Kieran Trippier's cross onto the head of Aurier as Tottenham banished memories of their Champions League defeat in style.
Dele Alli soon equalised and Son Heung-Min, who moved up-front, scored twice in the second half before Serge Aurier's first goal for the club added gloss to a victory which Spurs moved above Liverpool and into third place.
Pochettino said: "We played fantastic football".
And in another headache for Southgate, Dele Alli and Danny Rose also came off in the win over Bournemouth.
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Pochettino confirmed the injury is to the same ankle that kept Kane out for two months earlier in the season but they are hoping the problem won't be as serious this time around.
Spurs squandered a 1-0 lead to lose 2-1 to Serie A giants Juventus in the second leg of their Champions League last-16 clash on Wednesday night.
Bournemouth started positively and nearly took the lead in the fourth minute when Junior Stanislas broke through on goal and rounded the keeper, but he saw his effort come back off the post.
Bournemouth took control of the game after that as they went in search of an equaliser and came close in the 84 minute when Callum Wilson found the back of the net, but his goal was disallowed for a foul. I saw at half-time how they put pressure on the referee.
"It's a massive three points for us".
He volleyed in Alli's cross at the back post in the 62nd, before displaying his composure by rounding Begovic to seal three points in the 87th.