Those rumours are gathering pace with The Telegraph, The Guardian, The Express and The Sun all reporting that preliminary talks have been held with the Chilean's representatives.
According to the Telegraph, who run the story as their main sports lead: "A meeting is believed to have taken place in Paris last Friday at which it was discussed with Pellegrini’s representative whether the Malaga coach would be interested in taking over the club this summer."
The Malaga coach has a release clause in his contract of £3.5 million but that's unlikely to faze Roman Abramovich who has spent millions in compensation in both bringing in and sacking managers.
The one thing working against Pellegrini is his age – 59 – but given the high turnover of bosses at Stamford Bridge that again seems like a minor concern.
Pellegrini, who has also coached the likes of Real Madrid, Villarreal and River Plate in the past, has spent his entire career in Spanish-speaking countries but is fluent in English.
Roma and Manchester City are listed in The Sun as two other clubs who could be potentially interested in Pellegrini.
The Chilean has said that he is happy at Malaga, but financial problems at the club, which have led to a ban from Europe next season, could be the deal breaker.
Some papers, including The Times, have picked up on quotes from Spain, where the president of Real Madrid's youth program said that Mourinho has told them he will leave the club in the summer.
"Mourinho told us that he would not be able to come back to do the talk next year, as he will not be here," Manuel Alvarez, the Canillas president, told the Cadena Cope radio station.
"He said it jokingly and in seriousness, I do not know exactly which. He did not say where he would be going because we did not ask him any more. We only talked with him about whether he could get us a few tickets to Real’s Champions League game (against Borussia Dortmund)."
To transfer rumours and Newcastle's Tim Krul is on AC Milan’s summer hit-list, and the Italian giants are lining up a £12m bid. Goalkeeper Krul is out for the rest of the campaign with a shoulder injury that will require surgery next week, but that has not deterred Milan, who have already made enquiries about the Netherlands international. (The Mirror)
Manchester City misfit Roque Santa Cruz is in talks over a move to Brazilian club Botafogo. The 31-year-old has been enjoying a loan spell at Malaga and has also attracted attention from Flamengo. (Daily Mail)
Anzhi Makhachkala are preparing to offer Chris Samba a rapid return to Russia. The 29-year-old defender is expected to be one of the big names allowed to leave this summer, when QPR will try to slash their huge wage bill whether they stay up or not. (Daily Star)
And finally, Cardiff City's Malaysian owner Vincent Tan says the club could spend up to £25m on new players after securing promotion to the Premier League. (Telegraph)