It is understood that Wilshere, 24, made the call to go on loan as he wanted get back to playing regular first-team football having fallen down the pecking order at the Emirates Stadium during last season which was curtailed by leg injuries.
But when he returns to North London, Wilshere could be sorted out with a new deal as his current contract with the Gunners expires next summer.
An Arsenal source told The Sun: "Jack left because he wanted to get more first-team football to get his England spot back.
"He loves Arsenal and wants to remain and the club and the chats he has had with the club have been very positive on both sides.
"Jack is desperate to stay and prove himself to everyone what a top-class player he is. The experience about Bournemouth should serve him well."