The Northern Irishman only took over at the Scottish Premiership side earlier this summer.
The 43-year-old was appointed by the Hoops just over a month ago and is yet to take charge of his first game for the Scottish Premiership side.
Roy Hodgson announced his departure following England's Euro 2016 exit
Nevertheless, Roy Hodgson's departure following the Three Lions' humiliating loss to Iceland in the last 16 of the European Championships has seen the former Liverpool boss touted as one of the favourite candidates to take over.
But Rodgers has taken little time to distance himself from the position.
"I've just landed my dream job and I haven't even walked out for my first game," he told The Scottish Sun.
"I haven't put a foot wrong here! Brilliant. I see Jose Mourinho has also been linked with it. He's unbeaten too... Is it flattering? It's football. There are very few professions where one minute you are up, then down, then up again - that's just how it goes.
"If people think I can do jobs at a high level, then that's fine. But I'm busy here, worrying about what I have to do at Celtic."
The rumours linking the Northern Irishman with the post arguably proves that there are a lack of options who appear capable of taking the side forward. While Rodgers appears capable, there is arguably a sense of desperation linking someone who has taken another job so recently.
But it may also send a message to the Bhoys hierarchy that if they are not willing to back Rodgers, especially in the transfer market, his head may begin to be turned by the opportunities that are available elsewhere.