However, La Real’s sporting director Loren Juarros says the club have no plans to sell the “irreplaceable” Martinez during the current transfer window.
That would leave Barca having to pay the 26-year-old’s €32 million buyout clause in order to complete the signing this summer.
“It’s true Barca have spoken with us [about Martinez],” Loren said in a news conference on Friday.
“They’ve been in contact to show their interest in the player. But so far there is no offer on the table. That’s all we can say.
“La Real have no intention of facilitating Inigo’s exit to any club. He is irreplaceable. We do not want to sell him.”
Despite this, Martinez was left out of Real Sociedad’s squad for this weekend’s friendly against Watford, raising speculation that a sale was near completion.
Reports in both Basque Country and Catalonia say Martinez is set on moving to Camp Nou, with Diario Sport reporting the player has already agreed a five-year deal with the club.
Barca were not originally in the market for a central defender this summer, but new coach Ernesto Valverde has pushed for Martinez following the club’s tour of the United States.
Gerard Pique and Samuel Umtiti are expected to start the season as Barca’s first-choice pairing in the middle of the back four, with Javier Mascherano as their understudy.
Thomas Vermaelen is likely to leave the club, though, and Martinez’s arrival would also see Marlon Santos be allowed to go out on loan.
Barca have over €222m available to spend after Neymar bought out his contract on Thursday and completed a world record move to Paris Saint-Germain.
So far this summer they have signed winger Gerard Deulofeu and right-back Nelson Semedo, but they also want to bring in a forward to fill the hole left by Neymar and a central midfielder.