Saints are open to offers for the Poland international, who is also attracting interest from Italian club Napoli.
Bednarek has been at Southampton for five years, having moved to St Mary's from Lech Poznan in July 2017 for a fee of around £5million.
He has made 150 appearances for the south coast club in all competitions, scoring seven goals.
While the Pole is a regular starter under Ralph Hasenhuttl and one of their longest-serving players, the club are now prepared to cash in.
Napoli are keen to take Bednarek to Serie A but Leicester are also weighing up an offer to keep him in the Premier League.
The Foxes have yet to make a signing this summer despite Brendan Rodgers stressing the need to reinvigorate his squad.
Bednarek has three years left to run on his contract so is likely to command a decent fee.
Southampton to Leicester isn't a new route for players leaving St Mary's.
Last year the Foxes landed both Jannik Vestergaard and Ryan Bertrand from the Saints but they failed to make a huge impact at the King Power Stadium.