The Premier League title winner was unhappy with the terms that the Foxes had put on the table.
The midfielder turned down a contract that would have seen him pocket £80,000 a week, The Sun claims, and now the Foxes are planning to return with an improved offer of £100,000 a week.
The Sun adds that the length of time it took Leicester to initiate talks had left him questioning whether his future lied with Claudio Ranieri's side having been so influential in their title win last season.
The news that Drinkwater has turned down the first offer put on the table may well alert Tottenham Hotspur. The Sun reports that Mauricio Pochettino was keen on bringing him to White Hart Lane earlier in the summer, so hearing that talks have not gone smoothly may perhaps be enough to tempt Spurs to back up their interest with a concrete bid.
Tottenham manager Mauricio Pochettino may perhaps consider a move for the midfielder
If the 26-year-old is capable of replicating the form he showed last term, there is no doubt that he would have plenty to offer if a switch to North London took place.
He demonstrated that he has tremendous composure on the ball and is capable of playing some wonderful passes, and that would surely boost Spurs' hopes of improving on the promise they showed last season.
Leicester manager Claudio Ranieri will be hoping that the fresh terms are enough to get a deal done