According to a report in The Express, Pellegrini is looking to take advantage of Wilshere's expiring contract and move to sign the Englishman this summer.
Pellegrini has previously attempted to sign Wilshere when he was Manchester City manager between 2013 and 2016, and is apparently keen to finally land the midfielder when his contract expires shortly.
Despite making 38 appearances last season, Wilshere has struggled to hold down a position at Arsenal in recent years, settling mainly for games in the Europa League and EFL Cup. He was left out of Gareth Southgate's England squad for the World Cup this summer, and will be looking to prove himself next season.
After breaking through at Arsenal as a 16-year-old, expectations were incredibly high for the man who was compared to Xavi or Andrés Iniesta by former Barcelona defender Dani Alves. Unfortunately for Wilshere, a series of injuries have really prevented the 26-year-old from playing on a consistent basis.
This is not the first time that Wilshere has faced doubts around his physical conditioning. After an injury forced him to miss all but 3 matches of the 2015/16 season, he joined Bournemouth on loan for the 2016/17 season in an effort to prove himself to the world, including his manager at the time, Arsene Wenger.
He went on to make 29 appearances for The Cherries before yet another injury brought his season to an abrupt ending.
Wilshere was disappointed to be left out of Southgate's World Cup squad this summer, and will be desperate to prove himself to the world once again next season.