Ranieri was dismissed following his side's 2-1 Champions League defeat to Sevilla, just nine months after leading the Foxes to the most improbable of Premier League title successes.
The Italian defied odds of 5,000/1 to land the title for Leicester, but this season has been a different story, with the club lying just one point above the relegation zone.
Assistant manager Craig Shakespeare and Mike Stowell will take interim charge ahead of Monday's match against Liverpool, but former Manchester City manager Mancini, who had a short spell as a player with the Foxes and is available after leaving Inter Milan last year, is the favourite to replace Ranieri on a permanent basis.
Mancini said on Twitter: "I am sorry for my friend #Ranieri. He will remain in the history of @LCFC, in the heart of #Leicester fans and all football lovers."
Leicester chairman Vichai Srivaddhanaprabha, meanwhile, has urged angry fans to respect the decision to dispense with Ranieri and hinted at serious problems behind the scenes at the club.
The Thai businessman said the Foxes had "millions" of things to sort out if they were to "survive", describing the situation they were in as a crisis.
In a post on his Instagram account, Srivaddhanaprabha insisted to supporters that he had Leicester's best interests at heart and would "never let the club down".
In his statement, which was published on the BBC Leicester Sport Twitter account, Srivaddhanaprabha said: "Thank you to our followers who understand and still support me in any circumstances. What you have seen is only some sides of the club which we can show to the public.
"We have done our best as a management. We do not have only one problem to solve, but there are millions thing to do to make our club survive.
"For fans, who have done your best as the team supporters.
"I would like to take this crisis situation to thank you all fans, and at the same time I do understand you. I really appreciate for the fans who still (have) love and understanding. And also thanks for the ones who keep complaining to me and the management team. I do understand you too.
"Please respect my decision. I will never let the club down.
"Seven years of my hard work here, I make the club better and better in every way.
"No need to talk about money. As you can see. All money goes back to invest in everywhere for the club."
Srivaddhanaprabha also revealed he would no longer be posting on Instagram, apparently following demands by fans.
"I would like to take this opportunity to say good bye from this IG. You will not see me in IG anymore as you wish," he added.
