WASHINGTON—The Federal Bureau of Investigation suspects Russian intelligence agencies are behind the recent hacking of the emails of Hillary Clinton’s campaign chairman and of a contractor handling Florida voter data, according to people briefed on the investigations.

Top Russian officials on Wednesday, meanwhile, shifted away from denying a role in a separate hack of the Democratic National Committee. President Vladimir Putin said it is irrelevant who stole the computer records, and the foreign minister said the U.S. hasn’t...