The England Under-19 international underwent a medical on Friday.
Chambers, 19, has made 25 senior appearances for the Saints and Arsenal see him as an option at right-back, centre-back or in a more advanced role.
Arsenal have also enquired about Saints midfielder Morgan Schneiderlin, 24, who played for France at the World Cup.
In a BBC Sport feature on England's possible team for the 2018 World Cup, three of five national journalists questioned picked Chambers at right-back.
Chambers would become Arsenal's third summer signing following the arrivals of Barcelona forward Alexis Sanchez and Newcastle right-back Mathieu Debuchy.
He would be the latest player to leave St Mary's this summer following the departures of Rickie Lambert and Adam Lallana to Liverpool and Luke Shaw to Manchester United.
Defender Dejan Lovren is also set to undergo a medical with Liverpool after the clubs reached agreement on a £20m move.
And Tottenham have made an enquiry for Saints striker Jay Rodriguez.
Any confirmed Chambers deal is likely to see right-back Carl Jenkinson, 22, leave Arsenal on loan to gain regular first-team football.
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