The England U21 international has already had a medical and is set to become Arsenal's second new right-back of the summer following the signing of Mathieu Debuchy.
Chambers can also play at centre-half, where Arsene Wenger is looking to add depth following the departure of the versatile Bacary Sagna and ongoing Manchester United interest in Thomas Vermaelen.
And if the deal goes through the 19-year-old will become the latest first-team player to leave Southampton this summer.
Rickie Lambert, Adam Lallana and Luke Shaw have all followed manager Mauricio Pochettino out of the club, while Dejan Lovren is expected to finalise a move to Liverpool imminently.
Speaking after his side's friendly win over Bournemouth, new Saints boss Ronald Koeman told Sky Sports News: "Lovren said he is not in shape to play, but of course there is something underhand. We have to wait.
"I expect everything. I know the possibilities of some of the biggest clubs in England and Southampton had a great season. it's normal they could lose some players."
Morgan Schneiderlin has been linked with a move to Pochettino's Tottenham, and Koeman said: "He will start training on Saturday and then we have time enough to have a little talk."
As for new arrivals, Koeman again confirmed an interest in Aston Villa centre-half Ron Vlaar and said work was continuing behind the scenes on transfers.
Celtic's England goalkeeper Fraser Forster has also been linked, but Koeman said: "I don't say anything about names."