Begiristain was announced as the Premier League champions' new director of football on Sunday, becoming City's second recent capture from Barca after Ferran Sorriano joined as chief executive over the summer.
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And Vilanova, who recently succeeded Pep Guardiola in the Camp Nou hotseat after a spell as assistant manager, has highlighted the 48-year-old's role in the Spanish club's successes
"I congratulate Manchester City with the signing of Txiki, we must recognize the work he did here," he told reporters.
"Barcelona's achievements are his too, the titles we've won are, in part, thanks to his work with the team and with Pep Guardiola."
Begiristain, who brought the likes of Samuel Eto’o, Ronaldinho and Daniel Alves to Camp Nou during his tenure, will be tasked with leading City’s recruitment policy.
Goal.com also understands that the Premier League champions remain interested in Esteve Calzada, the ex-Barca marketing director who is also credited with playing an instrumental role in the club’s recent success.