United's co-chairman, Kevin McCabe, told the club website: "Nigel has a tremendous pedigree in the game. He brings undoubted experience and can provide stability, energy and confidence to the playing side."
Chris Morgan has been in temporary charge of the Blades for the last two games and guided them to a win over Port Vale and an impressive away draw at table-toppers Peterborough.
Morgan will continue to work as a coach alongside Clough, who takes over with the club sitting in the relegation zone on nine points after 13 games.
Clough's first game in charge will be at home to Crewe Alexandra on Saturday.