The former Arsenal midfielder was heavily linked with an Ibrox switch after his release from West Ham in October.
Wilshere has been training with Bournemouth since December and confirmation of his deal with the Cherries is to be announced today.
The 29-year-old, who played 197 times for Arsenal, spent a season on loan with Bournemouth in the Premier League when they were managed by Eddie Howe, who is now odds-on favourite to become the next Celtic boss.
He addressed the speculation in November saying the prospect of working with Steven Gerrard would interest him but no talks had taken place.
"It would interest me because it is Steven Gerrard, and that is no disrespect to Scotland,” Wilshere said.
"The door definitely isn't closed but, at the same time, it wouldn't be my first choice."