Drinkwater has emerged as a surprise target for Antonio Conte, with the Italian desperate to add depth to his squad after losing Nemanja Matic to Manchester United.
The champions have been linked with a £30m move for the 27-year-old in the hope of reuniting him with former Leicester midfielder N'Golo Kante and recreate the midfield combination that played such a major role in Leicester's Premier League triumph two seasons ago.
Last week, Drinkwater, who signed a new lucrative four-year contract said to be worth £90,000-a-week in August last year, reportedly told Leicester he wanted to move to Stamford Bridge.
However, Shakespeare believes the midfielder, a product of United's academy, will remain at the King Power Stadium this summer.
"We don't want to be a selling club, we want to add to it," Shakespeare was quoted as saying by the Daily Mail.
"We don't want to sell our best players. The speculation is always going to be out there at this time of year.
"You know when you have good players there might be an interest in them. You have to deal with reality but also keep a calmness about you as well."
Drinkwater will miss Leicester's first game of the season against Arsenal at the Emirates with a thigh injury but Riyad Mahrez could feature on Friday (11 August).
The Algerian winger has been linked with a move away from the Midlands throughout the summer, after submitting a transfer request earlier in the transfer window. However, despite being linked with Arsenal and Roma, Mahrez remains a Leicester player as both clubs' interest has petered out, largely due to the Foxes' financial demands.
The 2016 Premier League champions are understood to want in the region of £50m for Mahrez, who was named PFA Player of the Year two seasons ago.
Roma, who had at least two offers knocked back earlier in the summer, decided to pull out of the race for Mahrez after having a bid in the region of €30m turned down. However, Serie A rivals Inter Milan are understood to be unfazed from the price-tag and the club's director of football Walter Sabatini is believed to have made contact with Leicester over a move.
Earlier this week, Mahrez admitted his future was in his club's hands. "I know Roma came in, but nothing was accepted so there is little I can do," he said.
"They are a great club who I would like to talk to but I can't until Leicester accept a deal. Leicester know my thoughts, but I will continue to give my best for the club as I always have."
Despite the rumours surrounding the 26-year-old, Shakespeare said he had trained well and remained in contention to be included on Friday.
"I can only speak on how I've found him coming back, he has been focussed and driven," said the Leicester manager.
"He hasn't given me one problem where I've had to pull him to the side and say, 'Oi, pull your finger out.' Until we get a bid that is reasonable for Leicester, he will stay."