WASHINGTON—Government investigators have demanded that several bitcoin exchanges hand over comprehensive trading data to assist a probe into whether manipulation is distorting prices in markets linked to the cryptocurrency, according to people familiar with the matter.

The investigation followed the launch of bitcoin futures on CME Group Inc.’s exchange six months ago. CME’s bitcoin futures derive their final value from prices at four bitcoin exchanges: Bitstamp, Coinbase, itBit and Kraken. Manipulative trading in those markets...