The 24-year-old is being monitored by former Manchester United manager David Moyes, who is now in charge of the Spanish club as he is looks to convince the midfielder to return to his boyhood club.
With Sociedad tipped to continue their stay in the Spanish top flight, the Scottish boss is already believed to be plotting his summer transfer in order to rebuild his squad.
Illarramendi made 32 out of 38 starts in his final season with his former side before making a £24million switch to Madrid in the 2013-2014 campaign.
Although he made 15 starts in his initial campaign at the Bernabeu, he has been limited to only four first team appearances so far this year.
The two Premier League clubs were planning a deadline day deal to lure them to their respective sides, but the potential move fell through, making him a summer target instead.