The agency, which specialises in football finance, worked on the takeover deal that saw Swiss investor Markus Liebherr rescue the troubled club.
It is now understood Southampton wants a longer-term relationship with Square 1, under which the agency will advise on corporate PR.
This is likely to include representing Liebherr in the media in matters related to Southampton FC, as well as advising newly appointed board members chief operations officer Andy Oldknow and acting chief executive Nicola Cortese.
It is also believed that Square 1 will provide PR services to former West Ham and Charlton Athletic manager Alan Pardew, who will manage the team.
Square 1 Consulting chairman David Bick heads the account.
The firm has a long history of involvement in foreign takeovers of English football clubs. It handled US businessman Randy Lerner's acquisition of Aston Villa and deposed Thai prime minister Thaksin Shinawatra's controversial takeover of Manchester City. Square 1 also advised Dubai International Capital during its ultimately unsuccessful pursuit of Liverpool.