La Real is on the hunt for a new gloveman to replace Miguel Angel Moya, who terminated his contract at the end of June.
Brighton and Hove Albion keeper Ryan is currently on loan at Arsenal in the English Premier League, but made just three appearances for the Gunners last season.
He has 61 Socceroos caps and played previously in LaLiga with Valencia, with whom he made 21 appearances.
Reports suggest Real Sociedad is keen to complete any deal as quickly as possible, in order for Ryan to come into the side for pre-season training, which is scheduled to begin in the middle of July.
The club's aim is for him to create some healthy competition for the spot between the posts, with Roma goalkeeper Robin Olsen also rumoured to be a target and current keeper, Murcian Robert Sanchez, the third goalkeeper for Spain's national team.
There have also been suggestions Ryan could join Ange Postecoglou at Celtic, where the former Socceroos coach is looking to get he Hoops back on track after it gave up its Scottish Professional Football League title to bitter rival Rangers last season, Steven Gerrard's side winning the crown on the back of an extraordinary undefeated season.
Ryan has expressed interest in staying at Arsenal, but it is known that he wants more regular football than the Gunners were able to provide through the latter part of last season.
With crucial FIFA World Cup qualifiers beginning from September, a more active Ryan would also be helpful for Socceroos coach Graham Arnold, who has matches against Japan, China, Saudi Arabia, Vietnam and Oman to contemplate.
CONFIRMED: The Socceroos have been drawn into Group B of the #AsianQualifiers Final Round!— Socceroos (@Socceroos) July 1, 2021
The top two nations will advance to the 2022 FIFA World Cup while third place will enter the play-off rounds. #AllForTheSocceroos pic.twitter.com/iD5jHBe1ip
Parent club Brighton is also looking to reduce its wage bill, making a move elsewhere for Ryan more likely than not. The Seagulls would also welcome what transfer money it could get for the 29-year-old, whose contract is not up until 2022.
Should Ryan choose Real Sociedad and get selected in the first team, his first order of business will be to stop Lionel Messi and Barcelona in the opening match of the 2021-2022 LaLiga season on the weekend of 14-15 August.
Real Sociedad finished sixth in LaLiga last season, sealing a spot in the UEFA Europa League for 2021-2022.