Kellett has made just four senior appearances for Bolton and until three weeks ago he was on loan at Plymouth Argyle, who he made 15 appearances for in a three-month spell.
Kellett, 21, moves to Old Trafford as midfielder Saidy Janko, who joined Manchester United in September 2013 from FC Zurich, is set to go out on loan to Bolton.
Janko has yet to make a senior appearance for United, and the Swiss 19-year-old will now have the chance to push for first-team football with Bolton.
In Janko’s absence, Kellett’s siging will bolster the squad available to United’s U21 manager Warren Joyce.
Kellett is United’s third signing of the January transfer window, following moves for Sadiq El Fitouri and former Barcelona goalkeeper Victor Valdes.
Scottish midfielder??Darren Fletcher left United??to join??West Bromwich Albion on a free transfer.