Antonio Conte is determined to see three of four new signings arrive at Stamford Bridge before the summer window closes and Drinkwater is a top target.
Alex Oxlade-Chamberlain and Ross Barkley are also being linked with the Blues but Drinkwater is said to be keen on a move to west London to link up with the champions.
Chelsea have already seen a £25million offer rejected by Leicester City and the Daily Mail say a £30m bid has also been put on the table.
But the Foxes are in no hurry to sell the 27-year-old, who has impressed at the King Power since signing from Manchester United in 2012, and value the midfielder at £40m.
It is claimed discussions between the two parties are still ongoing but Leicester remain in a position of strength and Chelsea will be forced to dig deep if they are to secure a deal.
Speaking about speculation over a number of his players, with Riyad Mahrez also tipped to leave, Craig Shakespeare said: ‘I’ve read the speculation the same as you but until we get the bids accepted, there’s no difference for me.
‘We carry on with them. All the time we’re talking about the speculation. We’ll carry on as normal until I’m told otherwise.
‘I’ve said that your worst fear is that an acceptable bid comes in, but if it does, we have to deal with it. Until it does, we have no decision to make.’
Conte has spent close to £150m on Alvaro Morata, Tiemoue Bakayoko and Antonio Rudiger this summer but intends to bolster his squad before the August 31 deadline.
‘The club is trying its best in the transfer market to improve our squad but I must be happy,’ the Italian said after Sunday’s 2-1 over Tottenham.
‘We only have to think on the pitch and not to give answers to people.
Chelsea 2017/18 Fixtures
Everton (H) – Sun Aug 27
Leicester City (A) – Sat Sep 9
Arsenal (H) – Sun Sep 17
Stoke City (A) – Sat Sep 23
Man City (H) – Sat Sep 30
Crystal Palace (A) – Sat 14 Oct
Watford (H) – Mon 23 Oct
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‘This moment is not easy for us but I am very happy to have this group of players because they showed me great commitment, desire and will to improve after a great season.
‘We only need to focus on the pitch, outside does not matter for us.’