The Russian international, who is nearing the end of a six-month loan spell with the Whites, will be a free agent in the summer when his contract with Stuttgart expires.
Having made the decision to buy a property once owned by former Chelsea boss Mourinho, rumours have now circulated that the 28-year-old has agreed to extend his stay at Craven Cottage.
"I like my life in London very much," he told Russian website Tden. "Branislav Ivanovic, Ramires and Frank Lampard from Chelsea live in my neighbourhood as well as the players of Fulham.
"Jose Mourinho lived in my house before."
Pogrebnyak has scored five goals for Fulham.