The 25-year-old has been heavily linked with a move away from Estadio Jose Alvalade this summer, with a host of top European clubs believed to be keen on his signature.
Gunners boss Arsene Wenger has been credited with an interest in signing Patricio, as he looks to add competition for current number one Wojciech Szczesny at the Emirates Stadium.
However, Arsenal's reluctance to meet Sporting's £13million valuation of the Portuguese international means they now look to have lost out on the player, having had a £10million bid rejected earlier in the transfer window.
It's understood that money-bags Monaco have had an initial £9million offer for Patrico accepted, with a number of performance-related add-ons taking the deal far beyond the price that Sporting have been holding out for.
According to Correio da Manha, the Ligue 1 club will pay Sporting £2.5million once Particio plays 30 games for Monaco, and a further £8.5million if the club wins the French title during his time at Stade Louis II.
Additional extras based around the Champions League could see Monaco pay almost double the £13million that Sporting initially wanted for their goalkeeper, which completely blows Arsenal's highest offer out of the water.
If Patricio agrees personal terms with Monaco, he will join fellow countrymen Ricardo Carvalho and Joao Moutinho as part of Claudio Ranieri's new-look squad in the Principality for 2013/14.