The two clubs are in talks about a deal for the Netherlands international, who can play at centre-back and right-back, although there are still details to be worked out before a deal is reached.
Sources have told ESPN the fee for the 22-year-old is likely to be around €46.55 million ($50.75m).
Timber was heavily linked with a move to Manchester United last summer and although manager Erik ten Hag is an admirer, United are focused on other targets in the transfer window.
Timber signed a new contract with Ajax in August but sources have told ESPN that the Dutch side expected him to leave this summer amid interest from a number of Europe’s top clubs.
He is set to leave Ajax having made more than 100 appearances for his boyhood club following his breakthrough season in 2020-21.
He was part of title-winning teams under Ten Hag in 2021 and 2022 and has played 15 times for Netherlands after making his debut in 2021.
The arrival of Timber and Rice at Arsenal will take Mikel Arteta’s summer spend over £200m. Arsenal finished second in the Premier League behind champions Manchester City and are preparing for their return to the Champions League for the first time since the 2016-17 campaign.
Arsenal edge closer to signing Ajax defender Timber – sources