Midfielder Wilshere joined the Premier League club on a three-year deal on Monday after ending his long association with Arsenal, who allowed him to leave as a free agent at the end of his contract.
Hampered by injuries in the past, the England international made 20 league appearances in the 2017-18 season but now starts a fresh chapter of his career.
Wilshere, who will wear number 19, linked up with his West Ham team-mates at a pre-season training camp in Switzerland on Tuesday, with Pellegrini praising the 26-year-old’s desire to play for the club he supported as a boy.
“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 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.”
Wilshere is Pellegrini’s fourth signing since taking charge, following on from the arrivals of goalkeeper Lukasz Fabianski and defensive duo Ryan Fredericks and Issa Diop.
West Ham have also been strongly linked with Borussia Dortmund winger Andriy Yarmolenko, with the Ukraine international reportedly travelling to Switzerland to undergo a medical after a fee was agreed between the clubs.