The BT Sport presenter told TalkSport's Sean O'Brien that Spurs must keep the pair motivated to remain in north London by adding top-class players after Pochettino said he is not in charge of transfers.
"I can understand the frustration of the manager. He's done such an unbelievable job.
"I've been saying this about Tottenham for four years now, but this is a seminal moment for them and they've got to decide what they are.
"Are they just going to be a smaller club that punches above its weight occasionally? If they continue to do that, they will lose someone even as loyal as Harry Kane—because he'll want to win things.
"Sell, buy players, sign contract, not sign contract—I think it is not in my hands, it's in the club's hands and [chairman] Daniel Levy.
"The club needs to change my title and description. Of course I am the boss deciding the strategic play, but in another area I don't know. Today, I feel like I am the coach."
According to BBC Sport, upon signing a contract extension in 2016, the Argentinian boss had his role altered from head coach to manager by Spurs.
Spurs reportedly remain active in the transfer market before the Premier League window closes on Thursday. Paulo Dybala, Giovani Lo Celso and Philippe Coutinho have all been linked with moves to the club in recent days, per Rob Guest of Football.London. Tanguy Ndombele recently signed for the Lilywhites from Lyon for a club-record €60 million (£53.8 million).
Lineker added he thinks Kane would be content to commit his long-term future to Spurs, but if the club fail to sustain their progression, the England international could choose a move abroad, per O'Brien.
"I think Harry would love to spend his entire career at Tottenham, but I also think he's ambitious enough to want to win things.
"If it becomes a point where he sees Tottenham drift out of competitiveness, drop out of the top four and don't sign anyone, there comes a point where you consider your options.
"If he did leave, there would be a lot of clubs after him, including the giants abroad.
"From my personal experience, I hope Tottenham win things and he never leaves, but if he does leave it would be great to see him at Barcelona or Real Madrid."
Spurs' Premier League aspirations concluded prematurely last term, with the team finishing fourth. Pochettino's men had competed near the top for most of the season, but ended 27 points behind champions Manchester City.
However, the campaign ended with an appearance in the UEFA Champions League final. Spurs were defeated 2-0 by Liverpool, but qualifying for the final underlined how far the team has come under Pochettino. Spurs achieved this credible feat without making any signings last summer.