Solskjaer’s time at the Old Trafford helm remains under scrutiny following their 2-0 defeat at home to rivals Manchester City last time out.
The Red Devils now sit sixth in the Premier League table, nine points adrift of leaders Chelsea, with sections of the fanbase calling for Solskjaer to go.
United look set to stick with the former Molde boss for now but Leicester boss Rodgers remains on their radar.
However, according to Eurosport, Rodgers wants to see through the entire season with the Foxes before considering a potential move to United.
Rodgers doesn’t want to abandon his side midway through the campaign and believes they have a chance of further silverware in the Europa League this season to follow on from their FA Cup win last term.
United have been made aware any move they make for the 48-year-old in the coming weeks or months, should they sack Solskjaer, would likely fall on deaf ears from both Leicester and Rodgers.
However, that may not stop United expressing their interest with the club still confident they can tempt him away from the King Power Stadium.
The Northern Irishman was asked about the speculation linking him with the United job last week and replied: ‘I don’t even think about it. I’m sure the supporters are probably tired of it. At Leicester, there is always a link.
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‘My focus is always here with Leicester and developing the football club and challenging for honours. The directors, staff and players all have the same vision.’
United saw first hand Rodgers’ impressive work after suffering a 4-2 defeat to the Foxes last month.
The reverse fixture at Old Trafford doesn’t take place until April when Rodgers will expect to still be Leicester boss with his current contract running until the summer of 2025.
Solskjaer is currently at odds of 21/20 to be given the boot by December 26 or 5/6 to stay with Betfair.