But Cortese said the reports stemmed from a campaign designed to destabilise the club.
He said: "The club is not for sale. There's always been a group behind the scenes with an agenda that's not good."
Cortese, who conducted the deal to buy the club in 2009, added: "The commitment even after Markus passed away is exactly the same, and nothing has changed.
"This is a deliberate attempt to damage the club at this time because we're in negotiations with players and a very important [potential sponsorship] brand."
The future of Southampton has been uncertain since Liebherr passed away four months ago, with his daughter Katharina due to inherit the 62 year-old's empire. But Cortese insists the Liebherr family are not looking to offload the club.
"Southampton are in a very strong position and people envy that fact," he said.
It is essential to get promotion this season. I don't want to think of a scenario where that wouldn't happen
"If there's any lack of commitment [from the family] we have a plan B. I know people with money who would have the same attitude as Markus who would join us on this. But, just to underline, the commitment of the family is 100%."
Supporters remain fond of Liebherr after he rescued the club from administration in 2009, paying about £14m to save them, and Cortese says he is determined to achieve a legacy for the deceased magnate.
"I have been the boss of this from the beginning," he said. "Markus never made decisions, I did, and that is the same now.
"A few weeks after he died I was struggling with motivation, but it's not just about Markus now, it's for the people of Southampton and the legacy of Markus, I owe it to him. I'm driving his vision on."
Cortese is a former banker and remains friends with Italian Prime Minister Silvio Berlusconi, but denied rumours he is in line to take a directorship at AC Milan.
He said: "I'm here to stay. We're at the very beginning. I don't run away from things because something better comes along. We're really hoping to build something that will be massively successful in the future."
Adam Lallana is Southampton's top-scorer this seasonCortese controversially sacked former manager Alan Pardew, who is set to take over at Newcastle, in August, but said he had "no regrets" about the decision.
"I'm not going into details but there were plenty of reasons why it ended up with the sack," said Cortese. "All the attributes [current boss] Nigel Adkins has added is something we were missing before.
"He's a great communicator and motivator and I honestly believe we made a very good choice. I think I can have a very long-term working relationship with him.
"He's changed the attitude. His personality, his way of being, it's motivated people. People who haven't been here for a while, they come down here and they say 'it's a completely different atmosphere'."
Cortese believes the club, currently eighth in League One, are on track to return to the Premier League by 2014.
"We are actually ahead of schedule," he claimed "The plan remains the Premier League by 2014. I think it's realistic and we believe in it, our staff believe in it, the manager believes in it, so there's no reason why we shouldn't achieve it.
"It is essential to get promotion this season, to be in line with our plans. At the moment I don't want to think of a scenario where that wouldn't happen.
"I think we're in a very good position. We had a difficult start to the season but luckily the other teams are now beating each other. Now the teams ahead of us are closer and we're convinced that we're going to win this league."
Cortese added "he could not see any players leaving in January", including midfielders Adam Lallana, who has been linked with Birmingham, and Alex Oxlade-Chamberlain, who is being trailed by Arsenal.
"I can't see a possibility we would sell Adam in January," said Cortese, "and Alex is not going to go anywhere - we're talking about a very young player in the first year of a professional contract.
"We'll have to cope with the rumours and we have to help the kid cope with the rumours. He's very well looked after and I don't think he'd wish to be anywhere else than here.
"We're very proud to see the progress he's made in such a short period [but] there's still a lot for him to learn, and he's aware of that. We've got a great prospect for the future, but not the transfer market.
"There's a few players we're currently looking at, players to add value in League One, but players who can also do a job in the Championship. When we get into the Championship we'll want to be in the top six, already in year one.
"But even if you're offering big money in January and you get them interested, they've still got a big choice to make. That's why we can't make promises.
"We're strong candidates to do some business in January, but we won't do it for the sake of doing business. If not, we'll make another attempt in the summer."
Source: BBC Sport
Source: BBC Sport