There is believed to be a rift between Cortese and owner Katharina Liebherr, the daughter of the now-deceased Markus Liebherr who brought the Saints in 2009 and appointed Cortese.
While Cortese has handed in his resignation, it is believed the club have rejected it, although they are yet to make a public comment.
Cortese, a 45-year-old Italian banker, has overseen the South Coast club’s rise from League One and administration to the top half of the Premier League in just four and a half years.
Pochettino was appointed by Cortese in January of last year but said in May when it was rumoured the Italian would leave that he would only stay at the club if his chairman did.
The Argentine manager said: “If Cortese doesn't work here... I would not understand a Southampton with Nicola being here. The person who has put his faith in me at Southampton is Nicola Cortese. Of course I would not continue if he was not here.”
There has been a power struggle between Liebherr and Cortese and the former has tried to be involved on the board but Cortese has resisted the change.
For four months there have been discussions about the future of the club and the Italian has made it clear he is unhappy.
There have been reports in Italy that he could be looking to get involved with either Milan club or Roma but Sky Sports believe there is no truth in that, merely speculation.
Cortese’s tenure at St Mary’s has been an unmitigated success, with the Saints securing promotion from League One in 2011 with victory at Walsall on the final day of the season.
They made it back-to-back promotions under Nigel Adkins, finishing second in the Championship in 2012 before surviving with a 14th placed finish in the Premier League last season.
Adkins lost his job as Pochettino was appointed and this season the club have become even stronger under the Argentine.
Despite a poor run of form in December, Southampton remain in 9th place in the league and have no relegation fears.
Cortese has seen the club smash their transfer record three times during his time at the club, signing Jay Rodriguez for £6m in 2012 before Gaston Ramirez’s £12m arrival later that summer.
In summer 2013 Victor Wanyama and Dejan Lovren were both signed for large fees with Dani Osvaldo’s £13m deal breaking the record once again.
If Cortese does leave, and Pochettino follows him, the Saints will be in for a very difficult time.
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