The 55-year-old was taken to Cedars-Sinai Medical Center in Los Angeles on Monday, TMZ reported.
In a post on Instagram, he said: "I'm doing great and getting excellent care from my medical team."
He is "resting comfortably" after the aneurysm, his lawyer told Billboard.
In his post, Dr Dre also wrote: "I will be out of the hospital and back home soon. Shout out to all the great medical professionals at Cedars. One Love!!"
Friends and fellow stars have sent their well wishes after the reports of his ill health emerged.
Ice Cube, his former bandmate in trailblazing 1980s hip-hop group NWA, tweeted: "Send your love and prayers to the homie Dr. Dre."
Snoop Dogg, who was discovered by Dr Dre in the early 1990s, wrote on Instagram: "GET WELL DR DRE WE NEED U CUZ."
Missy Elliott wrote: "Prayers up for Dr. Dre and his family for healing & Strength over his mind & body." And singer Ciara tweeted: "Praying for you Dr. Dre. Praying for a full recovery."
With NWA and then as a solo artist, leading producer and record label mogul, Dr Dre shaped west coast rap and was instrumental in the careers of other stars like Eminem, 50 Cent and Kendrick Lamar.
An aneurysm is a bulge in a weakened blood vessel where the blood pressure causes a small area to bulge outwards.
Most brain aneurysms only cause noticeable symptoms if they burst, leading to bleeding on the brain, which can cause a very serious condition and can be fatal.