I've had a couple 'ITK' claims from a Spurs source. Maybe in line with what some may have heard. Summary of their claim:
Vlahovic shouts started around end of season, right when it became evident Kane wants gone. Spurs would have already had him, had Gattuso's appointment been finalised (you may recall that drama and also other issues with his appointment at Fiorentina only lasting 2 weeks and then at Spurs not going through). That was his guy, Paratici couldn't do that so in comes Nuno, and now they're wanting to pursue Vlahovic again, he was told he'd be the main guy and priority along with Gattuso. Vlahovic wants out of Fiorentina and his condition was he wants to work under Gattuso, now it's different, and Fiorentina are playing hardball over fee, around €60m. The brief from club is however any striker brought in will be to 'support Kane', some link Danny Ings as another one, obviously the smarter move, they want very little damage for when Kane leaves by showing off new signees.
In short, Spurs want a guy in before Kane is let go. The cash they get from us for Kane helps them offsets some immediate costs and get smaller deals across the line, they will obviously pay in installments over whomever they can get, so that's not an issue, just a matter of getting the right player.
imo, it would take a lot for this move to actually not go through rather than nothing happening and him staying. Obviously, as others have said, theres hurdles and U-turns, moves don't go through sometimes. These claims don't sound too wild for me, only hurdle would be if Levy decides to not go through on some of these planned signings, hold Kane against his will, and go into next season hoping for a top 4 finish, get some loan deals across the line. But I assume even he isn't that stupid (yet again, you never know).