Italian word fiume comes from Proto-Indo-European *bʰleh₁-, and later Proto-Italic *flowō (Flow.)
*bʰleh₁- (Proto-Indo-European)
*bʰley- (Proto-Indo-European)
*flowō (Proto-Italic)
Flow.
fluo (Latin)
I am soaked in. I flow, stream, pour.
flumen (Latin)
River.
fiume (Italian)