The Chilean, who guided Villarreal to the Champions League semi-finals in 2006, has taken debt-ridden Malaga to the quarter-finals in their debut campaign in Europe's premier club competition.
However, Malaga are banned from European action next season due to financial irregularities, and the 59-year-old has a buy-out clause of 4m euros in his contract, something Blues owner Roman Abramovich is ready to activate, the Daily Telegraph suggests.
Pellegrini has been on the shortlist for the role currently filled on an interim basis by Rafael Benitez, along with Everton boss David Moyes and Jose Mourinho.
Mourinho's Real Madrid future has been in doubt, but high demands, such as total control on transfer activity might scupper a return to Stamford Bridge after leaving acrimoniously in 2007.
Pellegrini does speak English, but hasn't won any major managerial honours since the now-defunct Intertoto Cup back in 2004.
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