The striker has been heavily linked with a move to City and even failed to report for training on the day he was due to make a return.
He won't come cheaply though, Spurs chairman Daniel Levy is understood to be holding out for £150million for his most prized asset.
No formal offer has been made by the Premier League champions, however, transfer expert Fabrizio Romano says this transfer saga is far from over.
Romano adds City aren't giving up on signing Kane this summer and will prepare an offer of €150m (£127.2m) and are willing to include add-ons if that's what's needed to convince Spurs.
He tweeted: "Manchester City are preparing their new bid for Harry Kane around €150m and would also include add-ons if necessary to convince Spurs."
"Tottenham still hope to keep Kane but Man City are not giving up - bid will be ready.
"Tottenham still hope to keep Kane but Man City are not giving up - bid will be ready."
The update comes after Chelsea announced they had re-signed Romelu Lukaku and talkSPORT pundit Trevor Sinclair that gives the Blues the upper hand in the title race.
To make matters awkward, Man City face Tottenham in north London on Sunday.
It's not clear if Kane will play and Spurs boss Nuno Espirito Santo kept his cards close to his chest on whether the 28-year-old will get the nod.
Asked about Kane being an option for the match, Nuno said: "All the players in the squad that work on a daily basis will be options for us, then it's up to us to decide."
He didn't criticise Kane over the timing of his return to training, but has placed emphasis on trying to resolve his future with the season about to begin.
Nuno added: "First of all we have to speak. This is going to happen today, and after that we hope to solve all the situations.
"It's not about that (Kane's late return), it's about what we have to do to solve the situation - this is important.
"We have to solve it together, everybody involved has to solve it together - speaking, having a nice conversation and putting our opinions, but everybody together."