Solskjaer has come under pressure in recent months following Manchester United’s poor start to the season.
The Norwegian, who is in his third full season at Old Trafford and is yet to win a trophy, brought in the likes of Raphael Varane, Jadon Sancho and Cristiano Ronaldo in the summer but United are languishing in sixth position after eleven Premier League games.
Manchester United have been outplayed and outclassed by Liverpool and Manchester City in their last two top flight home games, highlighting the gulf in class between the Red Devils and their rivals.
Rodgers has been linked with the Man Utd job should Solskjaer be sacked but the Northern Irishman says he is not interested in those speculations.
"There’s two things. Firstly, it is really disrespectful to ask the question when you have a manager in place, a good manager and a good man, who is working hard at the club,” Rodgers said ahead of Leicester hosting Chelsea in the Premier League on Saturday.
"Secondly, I can't really comment on it, because it is not something that is real.
“I am here as the Leicester City manager, proud to be here, privileged to be here and fully committed to the players, the club, the ownership. That’s about it, all the other noise around that is something we can’t control."
"It doesn't frustrate me because my focus is always looking for the next game. The only club I am thinking about is Chelsea and getting ready for that," the Northern Irishman said.
"It is frustrating for our supporters and also my players, if anyone is to read anything that is out there then that can destabilise.
"It is something that is the modern game, lots of speculation and gossip; it's all part of the modern game, but it’s not something I have any focus on."
Rodgers led Leicester to win their first ever FA Cup last season.
