The ex-Beatle joined other high-profile musicians including Madonna and Pete Townshend of The Who in calling for clemency in a case that has raised international concerns about Russia's commitment to free speech.
"I'm writing to show my support for you at this difficult time," McCartney said in an open letter released to the media.
"I would like you to know that I very much hope the Russian authorities would support the principle of free speech for all their citizens and not feel that they have to punish you for your protest."
He addressed the letter to Nadya, Katya and Masha, shortened first names of the three women on trial -- Nadezhda Tolokonnikova, 22, Yekaterina Samutsevich, 30 and Maria Alyokhina, 24.
They face up to three years in jail for their irreverent stunt in February, and the court ruling on Friday will be closely watched by the West for what it might say about Russian human rights under President Vladimir Putin.
The women have been held in jail since shortly after their performance, which offended many people in mostly Orthodox Christian Russia.
"I hope you can stay strong and believe that I and many others like me who believe in free speech will do everything in our power to support you and the idea of artistic freedom," McCartney added, concluding his message "Wishing you the very best of luck."
McCartney follows other top entertainers in calling for the women's release.
Townshend, along with Pulp singer Jarvis Cocker and Neil Tennant of the Pet Shop Boys, published a letter earlier this month to coincide with Putin's visit to London for the Olympic Games.
Madonna, performing at a concert in Moscow last week, donned a balaclava similar to those worn by Pussy Riot members and demanded they be freed to loud cheers from the crowd.
"I pray for their freedom," she said.
Small protests have been held in other cities around the world, and an informal group called "Free Pussy Riot!" has used social media to urge people to demonstrate on Friday.
(Reporting by Mike Collett-White; editing by Steve Addison)
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