The England midfielder, whose current Arsenal contract expires next summer, spent last season on loan at Bournemouth after being deemed surplus to requirements by head coach Arsene Wenger.
The 25-year-old missed the end of last season with a broken leg, but has since returned to fitness for Arsenal's Under-23 team.
Birmingham, in addition to a number of clubs, were strongly linked with a deadline-day move for the midfielder, who had been widely tipped to leave the Emirates Stadium before the transfer window slammed shut.
However, according to The Mirror, Wilshere rejected a £90,000-a-week deal at Birmingham as he was not prepared to drop into the Championship at this stage of his career.
Wilshere has been included in Arsenal's 25-man Premier League squad for the 2017-18 campaign.