Womack died Friday at age 70, though details about his death weren't immediately available.
He wrote the song It's All Over Now, which The Rolling Stones recorded in 1964. It was the band's first No. 1 hit on the UK Singles Chart.
The British band expressed its grief at Womack's death in a post on the Rolling Stones website.
"He was a true pioneer of soul and R&B, whose voice and songwriting touched millions. On stage, his presence was formidable. His talents put him up there with the greats. We will remember him, first and foremost, as a friend," the post read.