The former Liverpool and Celtic boss has impressed during his time at the King Power stadium, twice bringing them to the brink of Champions League qualification and winning the FA Cup last season.
The Northern Irishman was willing to walk away from the Scottish Premiership side in order to further his career back in 2019, and could now face another tricky decision.
With Michael Carrick taking over on a temporary basis while the club look for an interim until the end of the season, a big appointment is set to take place in the summer.
The Sun reported on Sunday evening that Rodgers had been contacted by United chiefs, who looked to have wasted no time in looking for a new permanent boss.
United are said to be willing to pay £8million to get their man from Leicester, which comes after a huge £7.5million pay-put for Solskjaer after he signed a new deal in the summer.
Rodgers is said to have a clause in his contract that allows him to speak to certain clubs if they show interest, one of which are said to be United.
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Rodgers said of his future in early November: “We’ve not been as consistent as what we have been. We can have a period of time now where we can get to work.
“My own commitment, I’ve got a huge respect for (chairman) Khun Top and the protection of his legacy.
“My relationship with him, (chief exec) Susan Whelan and (football officer) Jon Rudkin, we’re together and we’re looking to push on.
“I feel I’m in the right place. I work with a calibre of players we’re trying to maximise their potential. That’s why I’m so happy here.
“It’s interesting you talk about the top four, it shows the strides this squad has made. Our aim is to arrive in European football. There is still a long way to go, it’s still early in the season.
“My focus is on the next game.”