The Sun's Daniel Cutts exclusively reported as much, with both players said to be eager to remain at the Emirates Stadium beyond the end of the season, when their deals expire.
An unnamed source at the club said:
"Mesut loves it here. He loves living in London and has made it clear he wants to stay.
"Jack is Arsenal through-and-through and wants to become club captain next season. He wants to be a legend.
"They've both got the contracts on the table and there's not much else to iron out really. They should be all sorted soon."
Per James Olley of the Evening Standard, Arsenal boss Arsene Wenger recently stated the Germany international will not be sold in January:
While that could be because he does not wish to lose him and Alexis Sanchez in the same transfer window, it could equally be an indication there remains a possibility of tying him down.
The German has contributed four goals and five assists in 18 Premier League appearances this season. Though it has hardly been his best campaign in an Arsenal shirt, the standard of his play has been solid and generally higher than Sanchez's, whose form has dropped off a cliff compared to last year.
Meanwhile, Wilshere has successfully reintegrated into the team after breaking his leg while on loan at Bournemouth last year, recapturing some of the form that once saw him hailed as Arsenal and England's future.
Football writer Matt Spiro suggested renewing the 26-year-old's contract should be Arsenal's top priority:
ESPN FC's Mattias Karen has noted his importance to the Gunners on more than one occasion in recent weeks:
Given his injury record, a long-term deal could be seen as something of a gamble, even if Wilshere has managed the relatively impressive feat of completing 90 minutes in each of Arsenal's past seven Premier League matches, stretching back to the punishing festive period.
However, the deal's structure will mitigate the risk, per David Ornstein on BBC 5 live Sport:
It's a shrewd move from the Gunners, and if Sanchez ends up being the only player to leave, they would have done well to avoid a disastrous scenario of three key players departing in a short space of time.
There's plenty of rebuilding still to be done at the Emirates, but having Ozil and Wilshere around during that process should make it far smoother.