The 29-year-old Portugal international played under Marco Silva during his spell in charge of the Lisbon club and was linked to Watford earlier this year.
Back in March, Patricio admitted he couldn't be sure if he would finish his career with Sporting - where he has been for his entire career - leading to talk of a move to Watford.
The Hornets have been on the lookout for a long-term successor to Huerelho Gomes for a while now and don't appear to have found the answer with summer loan signing Oretsis Karnezis, who has been a disaster since he arrived from Udinese.
A move for Patricio, who is seven years younger than Gomes, could help to keep Hornets boss Silva happy at Vicarage Road.
Although the Portuguese coach now looks less likely to be tempted by the Everton vacancy, he is likely to be linked with other top jobs in the summer.