The Gunners were linked with signing the Bayern Munich defender on a loan deal this window but with Champions League qualification still a possibility, they may opt for Ake in a big-money move.
Arsenal would still prefer loan moves in January but are willing to make a permanent signing if it would help them secure Champions League qualification.
Bournemouth manager Eddie Howe is braced for bids for Ake after his side were drawn into a relegation scrap.
As part of the deal which saw the Dutchman move to the south coast, Chelsea have an option to re-sign Ake for £40 million.
This clause expires at the end of this window which could force the Blues to move for the centre-back.