The 25-year-old goalkeeper has been the subject of plenty of transfer interest, but with a contract at the Estadio Jose Alvalade that runs until 2018, the Portuguese giants are well placed to retain the player's services.
Gunners boss Arsene Wenger is believed to have identified Patricio as a potential successor to current first-choice stopper Wojciech Szczesny, who has recently been dropped by the French tactician amid concerns about his mental state.
Lukas Fabianski has been recalled to action, but with the Pole out of contract at the end of the season, there could be room for a new goalkeeper at the Emirates Stadium in the summer.
Manchester United boss Sir Alex Ferguson has also been linked with a number of different goalkeepers, as he seeks to increase competition for the number one spot at Old Trafford.
David De Gea currently occupies the first-team position, while recent rumours have suggested that Anders Lindegaard could leave the Red Devils at the end of the season, in search of more regular action.
However, if Arsenal or Manchester United hold any hopes of signing Patricio, Portuguese newspaper Record say it will take a lot more than £8.5million to secure his signature, bearing in mind the £34million release clause written into his contract.
It's understood that a representative working on behalf of the unnamed Premier League club, submitted an offer for the Portugal international that was swiftly knocked back by Sporting president Godinho Lopes.