Wilshere is set to fly to Switzerland to link up with new West Ham manager Manuel Pellegrini's squad at the club's pre-season training camp.
"It feels good, it feels special", said Wilshere, whose father and brother are also lifelong Hammers.
"I think it was a number of things", he said.
Personal terms between Anderson and West Ham are not expected to be a problem.
"At 26, he still has many years of football in front of him, and I believe he will be a great asset for us".
The 25-year-old Anderson is understood to be keen to play under Manuel Pellegrini, who was appointed West Ham manager in the summer.
"On the biggest stage, he has shown that he has game-changing abilities, as well as having the creativity and intelligence to make a real difference to our squad".
Wilshere made 197 appearances for the Gunners and helped them win back-to-back FA Cups in 2014 and 2015.
Among Wilshere's first objectives at his new club will be to play his way back into the England squad, having missed out on the 23-man panel for the World Cup in Russian Federation this summer. Wilshere is not to be the last player West Ham sign, as they are heavily on the trails of Andriy Yarmolenko of Borussia Dortmund and a few others.