The north London club confirmed the capture of the Brazilian 32-year-old winger on a free transfer on Friday.
Willian left Chelsea at the end of the season when his contract expired, ending his seven-year stay at Stamford Bridge.
The winger wanted a three-year contract while Chelsea were only prepared to offer a two-year deal; he has signed on for three years at Arsenal.
Thompson, who played in the Premier League for Liverpool, Coventry and Blackburn, feels that Willian has showed a remarkable lack of ambition by making the switch from Chelsea to Arsenal.
The former Red also feels Liverpool should have considered snapping up the winger as well.
Thompson wrote on Twitter: “Disappointed in Willian signing for Arsenal when you look at Chelsea’s ambition for next season.
“Surely [he] had higher aspirations than Arsenal.
“Surprised Liverpool didn’t have a bite for him, a truly gifted athlete even at 30 years’ old.”
Willian had offers in Europe and the MLS, but was convinced by Mikel Arteta’s vision for the club and Arsenal’s willingness to offer him a three-year contract.
