James Collins and Patrice Evra have both been released by the club after their contracts expired, while Sam Byram is also free to leave.
Pellegrini has also overseen the departure of four members of the backroom staff as the 64-year-old begins an overhaul at the London Stadium.
According to the report, two players that the Chilean is keen to sign are Manchester City's Yaya Toure and Chelsea goalkeeper Willy Caballero.
Toure is soon to be out of contract himself and like Caballero, worked under the management of Pellegrini for three years at City.
The duo are 35 and 36-years-old respectively, which like the signing of 37-year-old Evra in February, completely goes against what co-owner David Sullivan told the Guardian about the Hammers' transfer policy in December:
"We have to take a look at the age of the players we're signing. We will have to bring in two or three in January. They won't be old journeymen, they will be young players. They won’t be 32."