Portugal are on the hunt for a new manager after parting company with Fernando Santos following their quarterfinal exit at the 2022 FIFA World Cup in Qatar.
The 2016 European champions lost 1-0 to Morocco, bringing an end to Santos' eight-year stint in charge of the national team.
Mourinho joined AS Roma last summer and led the Giallorossi to their first ever European triumph when winning the inaugural UEFA Conference League.
The former Chelsea and Real Madrid boss is contracted at the Stadio Olympico until the summer of 2024 and El Shaarawy is optimistic he will not be swayed to take charge of his country for now.
"We have never talked to him about this possibility [of Mourinho leaving]. We think the coach is very focused on Roma, on the goals we have, on doing well here as we all are, as always," the winger told Il Messaggero.
"He is always present in every situation. Even in the way he talks to us in the locker room, in the match, in giving us the right indications.
"We see him serene, he is a person of the heart, instinctive, so the important thing for him now, I think, is Roma. We all hope he can stay here for a long time."
