The England forward, 25, is hoping to return to action later this month after picking up the injury against Manchester United in January.
But Lineker has urged Kane to err on the side of caution, fearing there is a deeper reason for his persistent ankle niggles.
Lineker told Goal: "It is always a worry when you get a recurring injury and there is clearly a little weakness in there.
"No player wants to have an injury that keeps cropping up, but ankles are not as bad as having a recurring calf problem or a hamstring issue, as they can be hard to overcome.
"You can roll your ankle and get away with it to an extent, but the more you do it, the more you worry the weakness will become a problem.
Harry Kane's ankle injury history
2018-19: Missed five games*
2017-18: Missed two games
2016-17: Missed three games
2016:17: Missed 11 games
*Correct as of February 9, 2019
"Maybe Kane can come back for the second leg of the Champions League tie against Borussia Dortmund.
"Having had this injury before, Harry will know when he is ready to play again."
Kane was set to miss two months with his latest setback but could be back in just half that time after the latest updates from Spurs.
Mauricio Pochettino is hoping to have his star striker back for the game against Burnley on February 23.
Since kicking off his senior career, Kane has suffered FOUR ankle injuries – totalling 21 games missed.
In fact, Kane has suffered one ankle knock this season – his torn ligament – with one last season and two the campaign before.
As Lineker suggests, it's a recurring theme and a potential worry for Kane, Spurs and England moving forward.
Tottenham will have to make do without the striker for tomorrow's game at home to Leicester.
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