WiMP, which rivals Spotify in some countries, is funded by paid subscriptions and has a one month free trial available.
It is currently available in Norway, Denmark, Sweden, Germany and Poland.
The service is also available in the US the UK, where it is called Tidal (which streams music in HD).
The move will put Jay-Z in competition with Beats Music, founded by Dr. Dre and bought by Apple in 2014.
Tidal currently offers users access to 25 million tracks, in addition to 75,000 music videos and other content including artist interviews for £19.99 ($24) per month.
That contrasts with Spotify’s charge of $12 per month for 30 million tracks.
Jay-Z has sold more than 100 million records worldwide and is estimated to be worth $520 million, but as well as being one of the world’s biggest selling music artists, he has a range of business interests.
If successful, Jay-Z’s bid will add streaming to a portfolio including 40/40 Club sports bars, Armand de Brignac Champagne and clothing brand Rocawear.
His company Project Panther described Aspiro as “an innovative high-quality company with strong future growth potential”.
WiMP has 512,000 paying users in Europe, making it far smaller than its main rival Spotify – it boasts 15 million paying subscribers across more than 60 countries.
The deal looks set to go ahead as Aspiro published a press release welcoming the offer.
“Panther is deemed to possess the capacity to develop the company in a privately owned environment,” the statement said.
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