The paintings were reported stolen in July 2015 along with a safe containing coin collections and jewellery and have yet to be recovered.
Spanish detectives were approached in February by British private investigators specialising in the recovery of stolen artworks who had received an email with photographs of the stolen works asking if they were listed as stolen, national police said in a statement.
Investigators then analysed the photo and were able to determine that the camera that took the images was owned by a photographic equipment rental company which supplied details of the customer who had rented it at the time the paintings were photographed.
The customer, who is suspected of involvement in the crime, was among those arrested, along with a Madrid art dealer and his son.
Sources close to the investigation said in March that the theft appeared to have been a highly-professional operation which took place while the owner was away in London, with the perpetrators disabling the alarm system.
The owner of the paintings was reportedly a close friend of Bacon.
Irish-born Bacon died in Madrid in 1992 aged 82 and his expressionist-surrealist works, which are often raw and emotional, remain hugely sought after.
Bacon's death only enhanced his reputation and the 2013 sale of his 1969 work “Three Studies of Lucien Freud” fetched $142,405,000 (€128 million) at auction, a world record at the time.
Art market information leader Artprice lists Bacon as one of 10 frontline modern artists alongside the likes of Picasso and Andy Warhol Warhol whose works make up 18 percent of global sales.