One song, Moonbeam Levels, was recorded during sessions for the album 1999 but was not included on the final LP.
A pretty, piano-led track, it touches on writers' block and an impending apocalypse. It will feature on the hits compilation Prince 4Ever next month.
The re-release of Purple Rain will feature an entire disc of outtakes.
According to Warner Bros and Prince's NPG Records, plans for the album "were agreed with Prince before he passed away" in April.
Details of the deluxe edition will be released at a later date, but they raise the prospect of an official release for songs like Electric Intercourse, Father's Song, All Day, All Night and Possessed, which have appeared on bootlegs from that era.
Longer versions of several Purple Rain album tracks also exist - including the title track, which contained an extra verse in the original recording.
Several full concerts from the Purple Rain tour were professionally recorded, including a March 1985 show that was released on a now-deleted home video.
Full details will be revealed "early next year", a press release said.
The greatest hits collection will be the first new Prince release since his death from an overdose of painkillers.
Among its 40 tracks are some of the star's best-loved hits including When Doves Cry, Raspberry Beret, Little Red Corvette and Kiss.
However it does not include any material released outside the star's contract with Warner Bros, which expired in 1996.
That means his sole UK number one, The Most Beautiful Girl In The World, and later hits such as Black Sweat and Musicology, are missing.
The advisers to Prince's estate, Charles Koppelman and L Londell McMillan, are currently considering proposals to license more material from Prince's legendary "vault", which is said to contain thousands of unreleased songs.
Follow us on Twitter @BBCNewsEnts, on Instagram at bbcnewsents, or if you have a story suggestion email entertainment.news@bbc.co.uk.
A new greatest hits album and a deluxe edition of Purple Rain will feature unreleased songs from Prince's "vault", Warner Bros Records has announced.