And the club’s officials have also complained to the Premier League about rivals Liverpool over their pursuit of £60million-rated Dutch international defender Virgil van Dijk.
Frenchman Puel is in the process of negotiating a pay-off over the remaining two years of his contract.
The Saints hierarchy have spent the past fortnight deliberating over Puel’s future following unrest from supporters and fallings-out with several senior players.
Vice-chairman of football Les Reed, who has led the inquest, was hoping Puel would save the club from forking out an expensive severance package by returning to his homeland with St Etienne.
But after holding talks with the Ligue 1 side, Puel decided he was not interested in the role.
Claude Puel took over Southampton last year after Ronald Koeman left for Everton Southampton's superb starting XI if they hadn't sold their stars Tue, June 6, 2017
Express Sport takes a look at what Southampton's team would have looked like had they not sold their best players
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How would Southampton's team look if they hadn't sold their brightest talents?
The former Lille and Nice coach only took charge at St Mary’s last summer when Ronald Koeman quit for Everton.
Saints finished eighth in the Premier League and were unlucky to lose to Manchester United in the EFL Cup final at Wembley.
But Puel’s negative tactics made him unpopular with the majority of fans and his constant tinkering also alienated players including Republic of Ireland striker Shane Long and Serbian Dusan Tadic.
Reed began searching for Puel’s successor even before the season had finished.
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Marco Silva was sounded out but opted to join Watford rather than wait for Saints to make their final decision on Puel.
Former Borussia Dortmund chief Thomas Tuchel is Saints’ dream target but would need convincing to make what would be seen as a step down from leading the German giants.
Tuchel’s fellow countryman and ex-Bayer Leverkusen manager Roger Schmidt is another contender to replace Puel, with Saints keen on making a swift appointment.
Saints, meanwhile, have responded to widespread reports that emerged this week that Van Dijk has indicated that he wants to join Jurgen Klopp’s club this summer.
But Southampton insist there has been no contact from their Premier League rivals and want the league to investigate an alleged illegal approach, direct to their player.
Van Dijk signed a six-year contract just 11 months ago, meaning Southampton have the upper hand in terms of negotiating his fee should they decide to cash in on their prize asset.
However, Klopp has made it clear that the Dutchman is his primary summer target and is also said to be ready to make him the best-paid player in Anfield history – with an offer better than the £150,000-a-week contract awarded to Philippe Coutinho last season.
But the latest news suggests that any negotiations over Van Dijk’s fee may not go smoothly. Southampton have to pay 10 per cent of the fee to Van Dijk’s former club Celtic because of a sell-on clause in the £11.5m transfer that brought him south two years ago.