The 37-year-old was officially unveiled as the new Saints boss last week following the departure of Ruben Selles, who left St Mary's at the end of last season following the club's relegation from the Premier League.
Martin, who spent two seasons in charge of Swansea City in the Championship, took his first training session as Southampton manager on Wednesday, but a number of his first-team players are facing uncertain futures at the club.
Indeed, captain and club legend James Ward-Prowse has been tipped to join a Premier League club this summer amid interest from the likes of Fulham and West Ham United, while Romeo Lavia has reportedly reached a 'verbal agreement' with Arsenal.
January signing Mislav Orsic is believed to be closing in on a move to Trabzonspor, while the likes of Romain Perraud, Paul Onuachu and Armel Bella-Kotchap have also attracted interest from elsewhere.
With Southampton potentially bracing themselves for a number of departures, Martin is tasked with recruiting the right players for his philosophy in an attempt to mount an immediate promotion challenge in the second tier.
According to The Sun, Martin is interested in reuniting with Swansea quartet Ryan Manning, Ben Cabango, Nathan Wood and Joel Piroe at St Mary's.
The report adds that Southampton would most likely move for just one of Cabango, 23, or Wood, 21, who have both established themselves as first-team regulars at centre-back for the Swans.
Manning is set to leave Swansea - who are now managed by Michael Duff - when his contact expires on June 30, but the Saints could face competition from Premier League duo Sheffield United and Luton Town, as well as Leeds United for his signature.
The 27-year-old left-back contributed with five goals and 10 assists for the Swans in 2022-23 and made a total of 88 appearances in two years under Martin.
As for Piroe, Swansea have a battle on their hands to keep their star striker who has scored 41 goals in 88 Championship appearances since joining the club two years ago.
The 23-year-old Dutchman is said to be firmly on Southampton's radar but they may face competition from fellow relegated side Leicester City.
Both Piroe and Wood have just a year remaining on their contracts at the Swansea.com Stadium, while Cabango's deal expires in June 2025.