Rangers boss Steven Gerrard played under Rodgers at Liverpool, where he narrowly failed to guide them to the Premier League title in 2014 before his departure in October 2015.
Gerrard, speaking on Tuesday, said: “Brendan has done ever so well at Celtic over the last few years. It is no surprise other clubs watch him and want to acquire his services.
“I have worked with Brendan, he is a very good coach. I am not surprised at the timing. From our point of view, we will keep an eye on it and see how it pans out. My focus is on Rangers.”
Puel was dismissed after 16 months in charge following Saturday’s 4-1 defeat by Crystal Palace.
The 57-year-old Frenchman left the club 12th in the Premier League having lost five of their past six league games.
First-team coaches Mike Stowell and Adam Sadler were put in temporary charge. Leicester play Brighton at home in the Premier League on Tuesday