Burke, 20, made 12 appearances for Wigan last season, while 21-year-old Cullen played 46 times for Bradford.
Both players join the Championship side having worked under Bolton manager Phil Parkinson at his former club Bradford.
"They'll add a lot of quality to the squad and have a desire to show everybody they can play at this level," Parkinson told the club website.
"It was an easy decision for me as I've worked with the gaffer before and I really enjoyed it," England Under-20 international Burke said.
Central-midfielder Cullen, a Republic of Ireland Under-21 international, helped Bradford reach the League One play-off semi-final and final in the last two campaigns.
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West Ham midfielders Reece Burke and Josh Cullen have joined Championship side Bolton on loan until January.