The Daily Record says Ancelotti, 58, has been contacted by the club and is their number one choice to replace his fellow Italian, Antonio Conte.
Conte is expected to leave before the start of next season - and could go earlier if things go badly against Barcelona in the Champions League last 16.
The former Italy manager has hinted at frustrations throughout the campaign.
And following last weekend's loss to West Ham he claimed that title hopes were over the season had begun.
Ancelotti clinched an FA Cup and Prem double in 2009-10, leaving the following year after two seasons at Stamford Bridge.