A spokesman for the reclusive German collector, who died Tuesday at age 81 at his apartment in Munich, said Gurlitt had living relatives but he would not say who they are.
It was also not immediately clear whether Gurlitt had written a will or whether a Munich court would appoint a curator of estate, which is often done in Germany if there are open questions surrounding an inheritance.
After much back and forth, Gurlitt eventually agreed last month to a deal with the German government under which hundreds of works he owned would be checked for possible Nazi-era pasts while staying in government hands. A spokeswoman for the Bavarian Justice Ministry told The Associated Press on Tuesday that deal would be binding on all possible heirs.
Initially, Gurlitt had insisted that all of the art work belonged to him and nobody else.
"Everybody involved — the authorities as well as private people who think some of the art may have once belonged to their families — wants to know more than anything what's going to happen to the collection," said Markus Stoetzel, a German lawyer specializing on the restitution of Nazi-looted art.
"The only thing we know for sure at this point is that the painful process of recovering art taken under Nazi terror will be further delayed," he added.
Gurlitt was thrust into the public spotlight in November when authorities, following a report by German magazine Focus, disclosed that they had seized 1,280 works by artists including Pablo Picasso, Henri Matisse and Marc Chagall from his Munich apartment more than a year earlier.
They had discovered the works while investigating Gurlitt for suspected import tax evasion.
Some of the pieces — by Matisse, Chagall and Otto Dix — were previously unknown, not listed in the detailed inventories compiled by art scholars.
Gurlitt had inherited the collection of paintings, prints, drawings and sculptures from his father, Hildebrand Gurlitt, an art dealer who traded in works confiscated by the Nazis and who died in 1956.
German authorities, facing criticism from Jewish groups and art experts for keeping the hoard secret for so long, quickly moved to publicize details of paintings online and put together a task force to speed their identification. They said at least 458 of the works may have been stolen from their owners by the Nazis.
Separately, representatives for Gurlitt later secured a further 238 artworks that were at a dilapidated house he owned in Salzburg, Austria. Gurlitt was never under investigation in Austria and those works weren't seized by authorities. It is not clear where those artworks are now.
Gurlitt stayed out of sight after news of his collection broke, barely talking to media, and was apparently overwhelmed by the publicity. In January, his representatives said they were considering Jewish families' claims for some of the works and said he was seeking "fair and just solutions" to the case.
"So much has happened in the past weeks and months, and is still happening," Gurlitt wrote on a newly created website shortly afterward. "I only wanted to live with my pictures, in peace and calm."
Gurlitt had undergone major heart surgery recently and returned to his home in Munich, where he had hoarded his art for several decades. He was "in nursing care and taken care of in recent weeks around the clock," his spokesman Stephan Holzinger said as he announced the death.
Gurlitt was born in Hamburg in 1932. He came from a prominent German family of artists, composers and collectors, but little is known about his life beyond his position as the heir of his father's art collection.
When U.S. investigators questioned Hildebrand Gurlitt after the end of World War II about the origins of his collection, they were doubtful whether all the pieces really belonged to him. Eventually, however, they decided that he was the rightful owner of most of them.
After his father's 1956 death in a car accident, Cornelius Gurlitt lived together with his mother in Munich, a former famous dancer, until she died in 1968. He reportedly lived a reclusive life, making a living by selling paintings from time to time.
Experts who examined the pieces seized in Munich said they included both what the Nazis called "degenerate art" as well as looted art.
The Nazis took so-called degenerate art — mostly avant-garde modern art, such as expressionism — from museums and public institutions because it was deemed a corrupting influence on the German people. Looted art was stolen or bought for a pittance from Jewish collectors who were forced to sell under duress during the Third Reich.
For the heirs of those collectors, the discovery of Gurlitt's massive hoard raised hopes of recovering some art, but the slow release of information by the German government has stirred frustration.
After the deal last month, German prosecutors who initially confiscated all the works of art they found at his apartment announced that they were releasing the rest of the collection. Gurlitt's lawyers had argued that the prosecutors acted disproportionately in seizing the entire collection, and that the art wasn't relevant as evidence for prosecutors' suspicion of import tax evasion.
Monika Gruetters, Germany's culture minister, said Tuesday that Gurlitt's decision to work with authorities deserved "recognition and respect."
"It will remain to Cornelius Gurlitt's credit that he ... sent an exemplary signal for the search for fair and just solutions with this avowal of moral responsibility," she said.
Geir Moulson contributed to this report.