Manchester City manager Pellegrini is considering a move for Brazil international Julio Cesar in January, reports Jason Burt of the Sunday Telegraph.
According to the article, the move will put pressure on England international Hart, who has been dropped by the Chilean in recent weeks in favour of Costel Pantilimon.
Rob Shepherd of the Mail On Sunday claims links between City and Barcelona star Messi should not be summarily dismissed.
Of more concern to City fans, though, is the link between striker Sergio Aguero and Real Madrid, as reported by Paul Brown of the Daily Star Sunday.
The article claims the Bernabeu giants are preparing a £45 million move for Aguero in January after deciding against pursuing Luis Suarez.
City striker Edin Dzeko has also been linked with a move away from the Etihad, with Arsenal and Borussia Dortmund monitoring his situation, claims Paul Hetherington of the Daily Star Sunday.
Pellegrini has also been linked with a move for Real defender Sergio Ramos, according to Tom Hopkinson and Dave Kidd of the Sunday People, while Ben Burrows of the Sunday Mirror claims Paris Saint-Germain midfielder Blaise Matuidi is the subject of City and Chelsea interest.