The guitarist was 67 years old, The Daily Mail reported.
Fast Eddie passed away peacefully on Wednesday in hospital, where he was being treated for pneumonia, the band said.
Clarke was the last surviving member of Motorhead's so-called classic lineup, which also included frontman Lemmy Kilmister and drummer Phil "Philthy Animal" Taylor, who died December 2015 and November 2015, respectively.
Clarke was introduced to Lemmy and Taylor in the mid-1970s and joined the band in 1976. In 2000, he made a guest appearance at their 25th anniversary concert at Brixton Academy, London.
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"The last of the three amigos", Dee wrote on Facebook.
Drummer Mikkey had only seen his friend recently and said he was in "great shape". He went on to form the two-piece hard-rock group Fastway with the former U.F.O. bassist Pete Way in 1983. His solos were short, sharp, belligerent affairs, nearly minimalist by design, but always firmly rooted in the purest essence of rock "n" roll, and as with all of the greatest architects of heavy music, you know an Eddie Clarke riff when you hear it. I was looking forward to seeing him in the United Kingdom this summer when we come around with the [Scorpions].
Our thoughts are with his family at this sad time. In 1973, he became lead guitarist in the blues prog rock band Zeus.