Before joining PSG, Rodriguez spent nine seasons at Real, scoring 12 times in 62 La Liga appearances. He won the Champions League in 2014 and 2016 before a knee injury stalled his progress.
"He hasn't had the happiest of periods in his career in Paris, but he's still only a young man and is hungry to make a big impression in the Premier League," Stoke chief executive Tony Scholes told the club's website (www.stokecityfc.com).
Rodriguez follows Darren Fletcher, Kurt Zouma, Eric Maxim Choupo-Moting, Bruno Martins Indi and Josh Tymon to the club.
Stoke, who host Arsenal in the league on Saturday, were beaten 1-0 by Everton in their season opener at the weekend.
(Reporting by Shrivathsa Sridhar in Bengaluru, editing by Larry King)
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