Earlier this week, the Gunners were reported to have entered discussions with their Dutch counterparts to sign the teenager, who is regarded as one of the most promising English goalkeepers.
Setford is yet to make his senior debut for Ajax, but he has made 33 appearances for their Under-18s and has also made the bench for the Under-21s, whom his brother Charlie Setford plays for.
The Haarlem-born shot-stopper has also risen through the England youth ranks and earned his first call-up to the Under-20s squad in June, making the bench for friendly successes against Sweden and the Republic of Ireland.
However, Setford's Ajax contract expires in the summer of 2025, leaving the Amsterdam powerhouses at risk of losing their academy star for nothing in under 12 months' time if a sale was not agreed in the coming weeks.
Setford 'to sign four-year Arsenal deal'
Following positive talks, The Athletic now reports that Arsenal have agreed a deal to bring Setford to North London, and the transaction will cost Mikel Arteta's side just €1m (£859,000) plus add-ons.
Setford will pen a four-year contract in North London until the summer of 2028, but he will be expected to join the club's academy ranks before making his first-team breakthrough.
A separate update from Fabrizio Romano claims that the 18-year-old will undergo his medical with the Premier League powerhouses next week, ending his lengthy association with Ajax.
Setford - who was born in the Netherlands to an English father - has spent over a decade in the Amsterdam ranks, having first signed up as an Under-8s player alongside his brother.
The 18-year-old is still eligible to represent both the Netherlands and England at senior international level, and he will become the third new goalkeeper to sign a permanent Arsenal deal this summer.
Significant changes in Arsenal goalkeeping department
Setford's arrival follows Arsenal's permanent capture of David Raya from Brentford following his successful loan last season, while Lucas Nygaard has also signed up to the club's academy.
The signings of Setford and Nygaard are a response to an exodus of goalkeepers from Hale End, as Arthur Okonkwo left at the end of his contract in June, while Ovie Ejeheri and James Hillson departed too.
A couple of shot-stoppers at senior level could also depart too, as third-choice Karl Hein - who has only just signed a new contract - is expected to be sent out on loan for the 2024-25 season.
Meanwhile, it is still unclear what the future holds for Aaron Ramsdale, who lost the number one spot to Raya last season and is expected to push for an exit in search of regular football.
In response, Arsenal made a bid for Wolverhampton Wanderers' Daniel Bentley - which was rejected - but they are expected to go back in for their former academy shot-stopper.