The 25-year-old is out of contract with the Gunners at the end of the season and had been tipped for a January exit, with the likes of Napoli and Real Betis interested, following his lack of game time in the early part of the campaign.
Arsenal boss Arsene Wenger, meanwhile, had some concerns over offering the midfielder a new contract due to his poor injury record, but he has now started their last three league games in a row.
According to The Sun, Wilshere's determination has convinced Wenger that he deserves to stay and has ordered negotiator Huss Fahmy to thrash out new terms.
The England international currently earns £90,000 a week at the Emirates and is said to be seeking £120,000 under his next contract.
Wilshere, who has been with Arsenal since the age of nine, recently revealed that he was "confident" of remaining with the club.