Both central defender Burke and midfielder Cullen are reunited with Bolton boss Phil Parkinson following loan spells at Bradford when he was in charge there.
“We’ve worked with Reece and Josh before and are very pleased to get them on board,” Parkinson told the club’s official website.
“They are good characters but more importantly good players. They’ll add a lot of quality to the squad and have a desire to show everybody that they can play at this level.”
Both players progressed through West Ham’s academy. Burke, 20, made 36 appearances for Bradford during a season-long loan spell in 2015/16 and featured 10 times for Wigan during a similar spell last season.
Cullen, 21, made over 60 appearances for Bradford during two separate loan spells and featured in their Sky Bet League One play-off final defeat to Millwall last season.