Antonio Conte's side have already had two bids rebuffed for the 27-year-old but have now been given a boost with Drinkwater asking to leave Leicester.
Last summer, the midfielder signed a new contract extension keeping him at the club until 2021.
The Foxes are determined to hold on to the man who played such a vital part of the title-winning squad in 2016.
They are holding out for an offer in excess of £30m, with talks between the two clubs still ongoing.
Blues summer signing Tiemoue Bakayoko would welcome Drinkwater's arrival. He told 'SFR Sport', "competition only makes you better. I am not scared of anyone."
"I think that there are not enough of us in midfield, especially with all the competitions we will be playing in.
"We need another midfielder to reinforce the team, so it is a very good thing that he could be coming."