The former Scotland international, who spent a brief period at Walsall in 2018-19, was expected to seal the move following the departure of Ruben Selles as head coach.
Martin will join from Swansea as the Saints move quickly to prepare for life in the Championship.
He verbally agreed to take over at St Mary’s over the weekend but the clubs remained in discussions over compensation.
Martin guided Swansea to 10th in the Championship last season despite having a bottom-four budget and the youngest squad in the division.
He has been praised for his style and Southampton will be hoping the 37-year-old can guide them back to the Premier League at the first attempt.
They will finish bottom of the Premier League after a disastrous season which has seen them use three managers.
Nathan Jones replaced Ralph Hasenhuttl in November but lasted just 14 games, winning one, and was sacked in February. Selles took charge for the final months of the campaign but was unable to keep Saints up.
On Monday, chief executive Martin Semmens stepped down from his role following relegation having been in the post since 2019.
The Saints have also confirmed Selles will leave the end of the season after talks.
A statement read: “Southampton Football Club can confirm that it has held conversations with Men’s First Team Manager, Ruben Selles, and decided that his contract will not be renewed when it expires at the end of the season.
“The club wants to place on record its thanks to Ruben for taking on the managerial position at a difficult time for the club and for giving his all as we attempted to stay in the Premier League.
“Ruben will take charge of the team for the final game against Liverpool on Sunday at St Mary’s. We wish him all the best for his future career.”