There have been widespread reports in the British media in recent days suggesting that the Blues are keen on luring the England international to Stamford Bridge following the sale of Nemanja Matic to Manchester United last week.
Chelsea are believed to be interested in landing the 27-year-old midfielder as Antonio Conte prepares for his second season in charge and gears his team up for a return to the Champions League next term.
However, it would appear that Leicester City boss Shakespeare has no intention of letting one of his club’s prized assets leave the King Power Stadium this summer.
Asked about the speculation linking Drinkwater with a switch to Chelsea, Shakespeare replied: “We don’t want to be a selling club, we want to add to it.
“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.”
Chelsea have been relatively active in the summer transfer window so far, having concluded deals to sign Willy Caballero, Antonio Rudiger, Tiemoue Bakayoko and Alvaro Morata.
Conte is currently preparing his Chelsea side to kick off their Premier League title defence with a home clash against Burnley on Saturday afternoon.
The Blues were beaten on penalties by Arsenal in the Community Shield at Wembley last weekend as they missed the chance to win their first trophy of the season.
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