According to Alex Crook of the Daily Mirror, Cherries manager Eddie Howe has made the pair his "top targets" for the upcoming window. Ake was on loan at the Vitality Stadium earlier in the campaign before the Blues recalled him to be part of their first-team squad.
Begovic, meanwhile, has found himself behind Thibaut Courtois in the pecking order at Stamford Bridge since signing in the summer of 2015. The 29-year-old has only made one start for the Blues in the Premier League this season.
While the pair may not have been first-team regulars under Antonio Conte this term, they have a role to play in the squad.
Ake, still just 22, could yet have a bright future at the club. The versatile defender has not featured frequently since returning from his loan spell with Bournemouth, but he played well in Saturday's 4-2 win over Tottenham Hotspur in the FA Cup semi-final, playing on the left side of a back three.
Sports journalist Paul Brown was impressed with the youngster's efforts, noting his contribution to a big win for the Blues:
Should he be guaranteed more minutes at Stamford Bridge you sense the Dutchman would be happy to stick around. But Ake's career is crying out for consistency.
Since 2015 he's played for four different clubs, having spent time on loan at Reading, Watford and most recently Bournemouth. He made 10 appearances for the latter and was a key figure in Howe's side starting the season strongly.
Begovic, according to Crook, is "keen to quit" Chelsea in the upcoming window, with Courtois cemented as the team's No. 1. The Blues have not performed particularly well with the Bosnia and Herzegovina international between the sticks either:
It'd be harsh to blame Begovic entirely for those woes, though. During his time with Portsmouth and Stoke City he was impressive and prior to his move to Chelsea, was rated as one of the best goalkeepers in the Premier League.
When he made the switch to the Blues there were concerns raised about whether he'd get the playing time needed to maintain those standards, however.
While he still has time to develop, another season on the fringes of the first team would also be a big setback for Ake.
For more news, rumours and related stories about Ake, Begovic, Chelsea and the Premier League check out the Premier League and Chelsea streams on Bleacher Report's app.