The England captain has amassed 217 goals in 328 games for the Lilywhites across all competitions, including 29 strikes this season, but he is yet to win a major trophy in North London.
Kane has supposedly informed Spurs of his desire to leave if they fail to qualify for next season's Champions League, and former Man United captain Keane believes that it would only be "natural" for the 27-year-old to consider a move elsewhere in order to fulfil his silverware ambitions.
Speaking to Sky Sports News, Keane said: "We obviously can't believe everything we hear, reading in the media, if he's actually said that.
"But it's only natural for us to speculate about Kane if they don't get the Champions League. Obviously he must be getting frustrated, obviously he's a huge player for Spurs, obviously he's a brilliant professional and a fantastic player.
"If they don't get in the Champions League - and they're not competing at the top level of the [Premier] League - then it's only natural that Harry Kane, from a selfish point of view, would have to look at his options.
"And he'll have some great options because he's a world-class striker and there are not many of them out there."
Manchester United and Manchester City have both been linked with a move for Kane, while Paris Saint-Germain manager Mauricio Pochettino is thought to have been in contact with his former striker over a summer transfer. body check tags ::