The left-back has been a regular in the La Real starting line-up for the past three seasons and started 35 of their 38 La Liga matches in 2016-17 as they finished sixth to qualify for the Europa League.
His displays attracted the attention of PSG boss Unai Emery and the 27-year-old looks set to swap San Sebastian for Paris during the close season.
La Real president Jokin Aperribay confirmed his impending departure on Thursday as the Basque club welcomed Diego Llorente from Real Madrid.
"Today a written offer for Yuri from PSG arrived and he is now allowed to undergo a medical," Aperribay said.
He returned to Spain in 2010 with Real Union, moving to Real Sociedad two years later.
Llorente, who made two Liga appearances for Madrid, has impressed in loan spells at Rayo Vallecano and Malaga in the past two seasons and has agreed a five-year deal at Anoeta.