City are among a cluster of clubs interested in holding talks with the 47-year-old German, who announced on Wednesday – as reported by Bild - that he will step down as Borussia Dortmund manager at the end of the season.
Klopp said he is "not tired" and "not planning a sabbatical" but "no longer feels he is the perfect coach for [Dortmund]", inviting interest from Europe's biggest clubs.
Representatives of Klopp have already received a number of enquiries for his services since news of his exit broke across Germany media outlets early on Wednesday.
Klopp has spoken in the past of his interest in managing in the Premier League, and City are among those keen to find out if he could be lured this summer.
Pellegrini's position is under threat in the wake of a disastrous campaign in which City will finish trophy-less and even face a late battle to clinch a top-four league finish.
City have been closely monitoring the situation of Carlo Ancelotti, who is likely to leave Real Madrid in the summer, and are understood to be willing to look at Klopp as a back-up option to the Italian.
The Premier League champions are keen on finalising their managerial situation over the next month and do not want it to drag on.
Speaking at an emotional press conference on Wednesday, Klopp said: "I have always said that the day I feel that I am no longer the perfect coach for this extraordinary club I will say that.
“That is something I have thought about in every phase here at Dortmund and decided in the last few weeks, days, that I was no longer able to be absolutely sure about that. And then, because of the unusual relationship I have with this club, the trust we have for each other, it was my duty to tell the club.”
Klopp arrived from Mainz seven years ago and has defied expectations to win the Bundesliga title twice, the German Cup and reached the Champions League final against Bayern Munich in 2013, a game they lost 2-1.
German newspaper Bild suggest that Thomas Tuchel, who replaced Klopp at Mainz, is the frontrunner to take over in the summer.