Citing sources, the Sun on Sunday says Ake, 22, is likely to sign a deal with Bournemouth when he returns from holiday next month.
The Netherlands international spent the first half of last season on loan at the Vitality Stadium and made 10 league appearances for Eddie Howe's side.
A source told the Sun that several clubs were interested in Ake but that Bournemouth were favourites to sign him.
"It should be done when the player gets back from his holiday next month," the source said.
"Others clubs have been taking a look at him, but Bournemouth look to be in pole position at present."
Ake joined Chelsea from Feyenoord in 2011. He only made two Premier League appearances for the Blues last season.
Chelsea boss Antonio Conte has been handed a £200m war chest to improve a squad that won the Premier League title last season.
The Italian coach is reported to have sanctioned a £100m outlay on Juventus defenders Alex Sandro and Leonardo Bonucci.
However, Juventus have no wish to part with Sandro and are willing to offer the wing-back a new contract until 2022 along with higher wages.
The Bianconeri are more open to selling Bonucci, who is understood to have a strained relationship with manager Massimiliano Allegri.
Conte is expected to sign a new contract at Chelsea worth £9.6m a year – more than the £7.5m a year paid to previous Blues boss Jose Mourinho.