According to Mundo Deportivo, the Basque club have been put off a move due to the hefty transfer fee Arsenal are requesting for the 32-year-old defender, whose contract in north London expires next summer.
Arsenal are not keen on letting one of their key players leave, hence the unspecified but apparently sizeable price tag.
Real Sociedad reportedly conducted a thorough search for a left back, with Monreal fitting their requirements perfectly as an experienced player who has played at the highest level with Arsenal. They also view him as a good potential mentor for their young defender, Kevin Rodrigues.
The Gunners signed Monreal from Malaga back in 2012, for a fee in the region of £15m. He has gone on to have an impressive career at the Emirates Stadium, and played 26 Premier League games over the course of last season.
Having recently returned from the World Cup with Spain, Monreal remains on holiday and is due to return to Arsenal for pre-season training next week.
Arsenal are set to continue their pre-season friendly schedule with International Champions Cup clashes against Atlético Madrid, Paris Saint-Germain and Chelsea.
They will then travel to Stockholm to take on Sevilla, before starting their Premier League season at home to Manchester City on August.
The Gunners have made five signings so far this summer, with the arrivals of Bernd Leno, Stephan Lichtsteiner, Sokratis Papastathopoulos, Lucas Torreira and Matteo Guendouzi.