The 31-year-old joined Mikel Arteta's side from London rivals Chelsea in January, penning a one-and-a-half-year contract until the summer of 2024.
Jorginho has since made a total of 30 appearances for the Gunners in all competitions, including eight starts and six substitute outings so far this season.
The Italy international is regarded as a valuable squad player by Arteta, but his long-term future at the Emirates Stadium is uncertain heading into the New Year when he can negotiate pre-contract agreements with foreign clubs.
Jorginho revealed last month that he is "really happy at Arsenal" and has no concerns over his future despite the lack of negotiations over a new contract.
The former Napoli and Hellas Verona midfielder then admitted earlier this month that he can see himself returning to Serie A, raising doubts over his Arsenal future.
Jorginho's agent Joao Santos has since provided an update on his client's future and has set Arsenal a deadline to negotiate a contract extension.
Asked whether Jorginho will renew his contract during an interview with TUTTOmercatoWEB, Santos said: "Everything is at a standstill.
"We'll think about it after February. However, the priority remains Arsenal."
Arsenal are understood to possess the option of extending Jorginho's contract by a further 12 months until the summer of 2025, and that clause could be exercised if the Gunners wish to retain their squad depth.
Jorginho is one of a number of Arsenal midfielders who is facing an uncertain future at the club along with injury-prone Thomas Partey, who could be allowed to leave the Emirates in January, according to a report.
Mohamed Elneny is in the same predicament as Jorginho as he will see his current contract expire in June 2024, while the likes of Emile Smith Rowe and Fabio Vieira have struggled for regular first-team football under Arteta.
Arsenal splashed the cash on two new midfielders last summer in the form of Declan Rice and Kai Havertz, and both players have featured regularly in the middle of the pitch along with captain Martin Odegaard.
Nevertheless, former Gunners midfielder Gilberto Silva recently revealed that Arteta's side are looking to bolster their options in their engine room during the January transfer window.
The likes of Aston Villa's Douglas Luiz, Al-Hilal's Ruben Neves, Fluminense's Andre and Manchester City's Kalvin Phillips have all been linked with a potential move to the red side of North London in recent weeks.