There could soon be a resolution, though, with leading suitors Bayern Munich to meet with Tottenham chairman Daniel Levy next week to try and strike a transfer.
It still won't be simple for Bayern, who have previously seen earlier bids of €70 million (£60m) and €80m (£68m) - both with add-ons - quickly rebuffed by Levy.
While talks over Kane are expected to intensify, Spurs will have been thinking about contingency plans if they are resigned to losing the England captain.
Such a plan could come as a replacement as a result of the large fee Kane will command or an option closer to home - someone already in manager Ange Postecoglou's squad.
Of course, directly replacing a player of Kane's ilk is an immeasurably difficult task, but Spurs must at least be exploring the ways in which that potential void could be filled.
Richarlison waiting in the wings
Starting with probably the easiest option, Spurs could thrust Richarlison out of his current second-fiddle role to Kane if the latter does leave.
Richarlison will be looking to put behind a frustrating 2022/23 campaign following his £60m move from Everton and that statement will ring true regardless if Kane stays or not.
The Brazil international netted just three goals last term - one of which came in the Premier League - with inconsistent runs in the team and injuries not exactly helping his case.
But in the first stages of Postecoglou's Spurs tenure, Richarlison already looks like a different player.
He scored a hat-trick in Tottenham's 5-1 win over Singapore side Lion City Sailors and had a menacing 45-minute cameo against West Ham the week prior.
Interestingly, and not to sound outlandish, there is some early evidence that Richarlison is better suited as the lone striker in the system Postecoglou deploys over Kane.
Kane is a typical centre forward and he has constantly shown how good he is on the ball when he drops deep, but Richarlison's aggressive press and mobility in behind seems to have clicked from the off.
Pre-season, granted, should usually be taken with a pinch of salt, but Richarlison looks hungry to prove a point and is staking his claim to play a much bigger role this season.
Splash the cash - but on who?
The other option for Tottenham would be to dip into the transfer market using any newly-acquired funds, and there are some intriguing names they could look to bring in.
Spurs would have the cash to stump up if they did want to bring in a Kane replacement, but the biggest questions are what type of player would they be after and how would they look to distribute their funds?
The latest name to crop up has been Randal Kolo Muani - a versatile forward who excelled for Eintracht Frankfurt last season with 23 goals in 46 appearances.
The Frenchman, 24, massively impressed in his first campaign with the German side and it is now no surprise that he is attracting Europe-wide interest this summer, including from Manchester United.
Kolo Muani has shown he can be a handy asset in front of goal, while his wide play using his pace is another string to his bow - something that could be well-suited to the way Postecoglou wants Spurs to play.
Then there is Dusan Vlahovic - a name that has long been linked with Tottenham which includes Antonio Conte's time at the club.
The Serbia international joined Juventus from Fiorentina last summer, but he hasn't slotted into the Old Lady's system perfectly.
Vlahovic is of a very different profile to Kane, who often likes to drop deep and dictate as much of the play as he can, whereas the Juve man is your typical target man and poacher in the box.
But that could work in Postecoglou's system given he sets up with a midfield three and attacking full-backs, who are there to carve out opportunities.
Brighton starlet Evan Ferguson could be the most like-for-like Kane replacement in terms of playing style, however, with the 18-year-old enjoying an excellent first season in senior football.
Ferguson, already a Republic of Ireland international, scored 10 goals in all competitions last campaign and his success came down to his rounded centre-forward profile.
Ferguson signed a new deal with Brighton in April this year and coupled with Brighton's recently tough negotiation style, that also makes him an expensive option.
There are plenty more options out there for Tottenham and they will have the funds available to acquire them if Kane does get his move, it's just a case of honing what they're really looking for.
Replacing a guaranteed 30-goal-per-season striker is not an easy job and in an ideal world for Spurs, they manage to keep Kane and later convince him to sign an extension, but that appears somewhat far-fetched given the stage he is at in his career.
