The 32-year-old, who was released by Newcastle at the end of last season, will move to the Estadio Riazor after seven seasons at St James' Park.
The former Argentina international overcame a battle with testicular cancer during his time in the North East.
He made an emotional return from illness in his final season at the club, sealing Premier League safety with a goal in Newcastle's 2-0 victory over West Ham on the final day of last season.
Gutierrez is expected to travel from his native Argentina to Spain in the coming days to team up with Victor Sanchez's La Liga side.