The Daily Mail claim that the Scot wants to stay in Spain.
The experienced Dutchman came in mid-season in 2014-15 and kept the Stadium of Light outfit in the top flight, but this campaign has been dire so far for the North East club.
With Advocaat no longer at the club, Sunderland are eager to make a managerial appointment during the international break and reportedly had Moyes in their sights.
The Daily Mail claim that the former Everton and Manchester United boss has rejected the chance to return to English football, with his quest with Real Sociedad not complete.
The San Sebastian club were beaten 3-1 by Malaga in their last game and sit 16th in the La Liga table after seven fixtures, with only one win to date for Moyes and his men.