Here, Press Association Sport looks at five possible names in the frame.
The Portuguese has been widely mooted as a possible replacement over recent weeks and is available having left Tianjin Quanjian in the summer. Well known to English fans after spells with QPR, Swansea and Leicester, the 48-year-old has coached – and played – all over the world and has a silky style which may cheer Saints fans.
The go-to rescue act when clubs are in crisis, this would appear, on the face of it, the perfect job at the perfect time of the season for the former England manager. He has successfully saved Sunderland and Crystal Palace in recent seasons, while stabilising Everton last year before criticism of the team's style kicked in. Now 64, would he fancy another mission?
Sacked by Fulham last month after taking them into the Premier League, the Serb will still have many admirers and is now available, having done good work at Watford, Maccabi Tel Aviv and then at Craven Cottage. He would be an interesting choice, but would a return come too soon?
Sacked by Monaco earlier this season, Jardim represents another possible option for the Saints. Has lots of experience and has won titles in Portgual, Greece and France, most notably when beating Paris St Germain to the Ligue 1 title in 2017.
A former Saints player, Monk is rebuilding his managerial reputation with some style at Birmingham after a short-lived spell at Middlesbrough. Did excellent work at Leeds in 2016/17 and was considered unlucky to lose his job at Swansea prior to that, having kept them in the Premier League. Two members of his coaching staff, Darryl Flahavan and James Beattie, are ex-Saints.