The 45-year-old Italian banker, who became the Saints' executive chairman in 2009, has suffered an "irretrievable breakdown" in relations with club owners the Liebherr estate, the Southern Daily Echo said on Wednesday.
Sky Sports reported that Cortese had already submitted his resignation amid persistent reports in the British and Italian media saying AC Milan want to lure him to the San Siro.
"Sources close to the top of the club have today indicated that Cortese's departure could come within a matter of days rather than at the end of the season," the Echo added.
Cortese's future was the subject of speculation last summer before an interim agreement between him and Liebherr estate over the direction of the club as well as his contract.
But, the Echo reported, discussions since then to agree a long-term strategy have not produced a solution, and Cortese is considering his position at St Mary's Stadium.
His departure would be a major blow to Southampton, not least because Argentine manager Mauricio Pochettino, 41, suggested last May that he would also quit if Cortese left.
Bookmaker William Hill said on Wednesday they had seen a surge of bets on Pochettino to leave the Saints, and had installed him the 7/4 favourite -- in from 14/1 -- to be the next Premier League manager to part ways with his club.
"The odds suggest that Mauro Pochettino could be leaving his role imminently," William Hill spokesman Rupert Adams said.
Cortese persuaded German-born industrialist Markus Liebherr to buy the club in 2009, when Saints were in administration and playing in the third tier. Liebherr died in August 2010, when ownership of the south-coast club passed to his family estate.
CONTROVERSIAL ACT
Cortese ran the club's day-to-day affairs as Southampton climbed from League One to the Premier League in three years.
His most controversial act was to sack manager Nigel Adkins in January 2013 and replace him with Pochettino.
Fans were highly critical of the change and the treatment of Adkins but former defender Pochettino has won them round, ensuring Southampton retained their top flight status last term with wins over Manchester City and Chelsea along the way.
Cortese has also been a shrewd operator in the transfer market, backing his managers to bring in now-established players such as Rickie Lambert, Jay Rodriguez, Victor Wanyama, Dejan Lovren and Dani Osvaldo, who arrived from AS Roma in August 2013 for a club-record fee of 15 million pounds ($24.69 million).
When rumours of Milan's interest first surfaced in 2010, Cortese told Southampton's website (www.saintsfc.co.uk):
"I am obviously flattered about the interest, and flattered that it has come from a top, top team because I think this is a success, not just for me, but for Southampton Football Club, the supporters, my management team and our first team.
"My answer at the time when I had this approach was a simple one and I didn't have to think for even a second about it. The club (Milan), despite where they are and the success that they have had in the past in the Italian league, cannot offer me anything that Southampton cannot achieve."
(Writing by Stephen Wood; Editing by Ken Ferris)
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