The Hammers are currently closing in on their new head coach after David Moyes left the club last week, with Pellegrini tipped as the favourite to take over this summer ahead of Newcastle boss Rafa Benitez.
The former City manager could waste no time in rebuilding the squad this summer, with the Telegraph reporting that West Ham could have the upper hand in the race to sign Touré, thanks to his previously strong relationship with Pellegrini.
The pair worked together at the Etihad for three years, with Touré cementing his place as a Citizens legend when the two lifted the Premier League title together in 2014.
A reunion could be on the cards next season if Touré opts to stay in the Premier League this summer, in a move which would extend his eight-year spell in the English top flight.
Touré has been linked with moves all across the world following his release from City, with many expecting that the 35-year-old would move to either China or America.
This latest report seemingly affirms Pep Guardiola's belief however, that Touré still has something to offer clubs in the Premier League next season.
"I am not the right guy to say to him what is the best. He will choose," Guardiola said, as quoted by The Sun.
"I think he has still the desire to play. He loves to play football. He likes the game and he enjoys the training sessions, especially with the ball. My advice, if he still has the desire to play, he has to continue. If you have doubt, you have to retire, it is over."
Toure's potential arrival at the London Stadium could see the Ivorian regain his place as a Premier League regular, with the Hammers keen to improve their star quality in central midfield.