The Hungary international joined the Cherries from AZ Alkmaar two years ago and he has already established himself as a key player.
Kerkez has eight goal contributions from left-back this term, and reports claim that this could be his final campaign at the South Coast outfit.
Chelsea and Liverpool are eyeing upgrades in the left-back department. The Cherries are prepared to sanction his sale for around £45 million.
Kerkez could be one of many players heading out of Bournemouth.
Dean Huijsen has also been linked with Chelsea, Liverpool and Real Madrid. He has a £50 million release clause in his contract.
Antoine Semenyo could also head for the exit door. Al Nassr are reportedly preparing a huge £75 million bid for the Ghana international.
There is also speculation that Justin Kluivert could contemplate a new challenge elsewhere if the Cherries don't qualify for Europe.