The Leicester boss is seen as one of the candidates to take over from Ole Gunnar Solskjaer but he rejected the speculation at a press conference.
Rodgers said: "It's not real. There's always gossip and that will be there until they appoint a full-time manager.
"My commitment is to here. The culture here is about the development of the players and that's where I see it, even more so in this period.
"I respect you guys have questions to ask, but that doesn't include me."
Rodgers was speaking as Leicester prepare for their Europa League match at home to Legia Warsaw, where they could be knocked out of the competition if they lose to the Polish side.
Leicester's European struggles have formed part of an inconsistent season so far but Rodgers called from the club's supporters following success in recent years.
The coach said: "I understand frustrations but I just ask for a bit of perspective.
"We have to be together. The King Power Stadium is a really difficult place to play when there is the synergy between the fans and players. Bring that incredible passion and support to help the team, and we need that, because we need to win to jump up the table. We need them - and for us, rather than against us."