The 26-year-old completed his move across London on Monday, a week on from his Arsenal contract expiring, and brings plenty of Premier League experience with him.
Wilshere has had to contend with numerous injury setbacks throughout his time with the Gunners, but Pellegrini is hopeful that the Englishman can put those problems behind him and inject some energy into United's midfield.
"We are very happy to welcome Jack to West Ham United and I look forward to working with him, starting at our training camp here in Switzerland," Pellegrini told the club's official website.
"Jack is a player with excellent technical ability, combined with great energy and commitment, and has proven his quality both in the Premier League and at international level," he told the club's official website.
"Most importantly, he is hungry to play for West Ham, and determined to reproduce the level of performance that he has shown in the past. At 26, he still has many years of football in front of him, and I believe he will be a great asset for us."