The Daily Mail now links former Borussia Dortmund manager Jurgen Klopp with the job but also state that former Chelsea boss Carlo Ancelotti could also be in contention.
Rodgers has come under increasing pressure in recent weeks after a series of poor performances saw Liverpool take just two points from its past four English Premier League games.
The Reds progressed 3-2 on penalties after the match had finished 1-1 after extra time.
Asked about his thoughts on whether his job was under threat he said: “I’ve never complacent enough to think that it’s never been”. If you’re not winning, that will intensify. It’s massive but you approach it the same way with the same mentality and same style as any other game.
“I have regular contact with the owners and relationship has always been strong”. If you don’t get results the pressure is there but I enjoy challenges.
‘He would have won the Premier League for Liverpool for the first time in God knows how many years.
“I expect to be a manager for another 20 years, and I don’t expect that those 20 years will be here at Liverpool”.
“The players that are selectable really want to play and fight for the team”.
Rodgers managed to keep his place during the summer despite a disappointing 2014-15 campaign, where he failed to lead his side to the Champions League.
Liverpool-based journalist Graham Beecroft made the claim in an interview with radio station Talksport, and says the Reds hierarchy came to the decision prior to Wednesday night’s narrow win over League Two outfit Carlisle United.
Rodgers, whose side scrapped through their Capital One Cup tie with Carlisle on penalties, returned to work as normal on Thursday and was the last senior member of the club’s coaching staff to leave their Melwood training base shortly before 6pm. “What drives a manager on is that fear, a positive fear, that every day you need to be at your best.” he said. I’ll walk away and hopefully I’ll get into another job where I’ll get the chance to create something special.
“I’ve an inherent belief in what I do”. We almost achieved great things. I respect that we need to get results. But to work with the players I’ve worked with here has been fantastic.