The singer died in April from an accidental overdose of the painkiller fentanyl, but he did not leave a will and did not have any children.
Reports in Minnesota say his estate may be worth at least $300m (Â£227m).
A judge has now ordered genetic tests to be carried out on six people with claims to be relatives of the singer.
Four siblings or half-siblings, as well as two women believed to be a niece and a grand-niece, will be tested.
Genetic tests had already ruled out the claim of a man in jail in Colorado who said he was Prince's son.
And, as part of the latest ruling, Judge Kevin Eide decided a claim by a Georgia woman, who said she and Prince were once married, was not valid. She had said the CIA was keeping their marriage record secret.
Minneapolis' Star Tribune newspaper said the case was thrown out against five men who said Prince was their father.
Prince was found unresponsive in a lift at his Paisley Park Studios on the morning of 21 April, local officials said. First responders tried to revive him with CPR but he was pronounced dead shortly afterwards.
Prescription painkillers were in the 57-year-old's possession following his death, officials told US media in May.
He is believed to have suffered from knee and hip pain from years of performing.
According to the autopsy report, Prince self-administered fentanyl, an opioid many times more powerful than heroin.
A singer, songwriter, arranger and multi-instrumentalist, Prince recorded more than 30 albums. His best known hits include Let's Go Crazy and When Doves Cry.

BBC Sport will increase its TV coverage of all 12 international events, which will also be covered on the BBC Sport website and BBC Radio 5 live.
"The deal strengthens the BBC's long-term commitment to athletics," said BBC director of sport Barbara Slater.
"It is our biggest commitment in terms of coverage of the Diamond League on network television."
The BBC has a separate deal for live coverage of the two UK Diamond League events in Birmingham (7 June) and the Anniversary Games in London (24 and 25 July) which runs through to 2020.
Lamine Diack, the International Association of Athletics Federations president and chairman of the board of the Diamond League, said: "We are very proud to have a partner like the BBC to commit to such a broad coverage of our meeting series.
"BBC's support is of course also great news with regards to the upcoming IAAF World Athletics Championships in London 2017 and confirms the importance of our sport in the UK."

The images are all the more striking when you consider that Mr Jenkinson, 42, from Y Ffor, near Pwllheli in Gwynedd, is blind.
He suffers from panhypopituitarism, a condition which has left him with just 5-10% vision in his left eye and 50% in his right eye. He has no perception of 3D and is also colour-blind.
Nevertheless, he is a keen photographer who works hard to capture "the best shot I can".
Mr Jenkinson, who was diagnosed at the age of 28, said doctors think the condition was caused by a piece of his skull not forming correctly in the womb.
This allowed fluid from his brain to squash his pituitary gland and his optic nerves.
"The last 10 years or so have been crushing, medically," he said.
"I was put on hormone therapy and have been taking steroids for 14 years but a side effect of that is that I now have osteoarthritis in both knees and hips."
An operation to try to correct his vision was not successful and temporarily left him with seizures. He has also suffered a "severe" aneurysm and 18 months ago, he was diagnosed with prostate cancer.
A former building control officer for Gwynedd council, Mr Jenkinson's health problems have meant he has not been able to work for a decade.
"I have had some very, very low days after my cancer diagnosis, even some suicidal moments," he said. "But my photography helps me cope.
"When you're concentrating on something, like how to get the best shot or where to set up your tripod, you're not thinking about your problems.
"I hope my story will help others in the same situation to know that there can be some light at the end of the tunnel, as I have found in my photography."
A judge in the US state of Minnesota has dismissed claims by 29 people who said they were owed a share of Prince's inheritance.
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Highlights of the 2016 and 2017 Diamond League seasons will be available to watch on the BBC as part of a new deal.
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From Afon Dwyfor to the animals in the Welsh Mountain Zoo, Paul Jenkinson's pictures show his native north Wales in all its glory.