The 22-year-old joined the Red Devils in 2011, and after spending two years in the youth system at Old Trafford, made his first-team debut during the 2013-14 campaign.
Januzaj impressed in his maiden season for Man United but since then he has fallen out of favour at the club, spending the 2016-17 campaign at Sunderland.
Just one year remains on the Belgian's contract at Old Trafford and, according to Marca, Sociedad are hopeful that Jose Mourinho will allow Januzaj to leave on a permanent deal.
The report goes on to say that La Real president Jokin Aperribay believes a transfer is a realistic premise although Man United are believed to want around £9m for the player.
Januzaj, who has also been on loan at Borussia Dortmund, failed to score at the Stadium of Light as Sunderland finished bottom of the Premier League table last season.