The move sees both players reunited with Bolton boss Phil Parkinson, who also took them on loan during his time in charge of Bradford City.
Burke spent last season at Championship side Wigan Athletic, while Cullen helped Bradford to the League One playoff final.
"It was an easy decision for me as I've worked with the gaffer before and I really enjoyed it," the 20-year-old Burke told Bolton's official website.
"He's a good coach and I worked well with him and the coaching staff. I think I enjoyed one of my best seasons with them before. Hopefully they bring out the best in me this season as well."
Cullen added: "It's brilliant to be here and I'm raring to go. The gaffer is a big part in me coming here. I got good experience with him at Bradford and feel I improved as a player.
"I'm now coming to a Championship club and looking to make the next step. Hopefully we can kick on and go on to have a successful season."
Parkinson was also happy to be reunited with the duo and is confident that they will prove to be important players during their first season back in the Championship.
"We've worked with Reece and Josh before and are very pleased to get them on board. We'd like to thank West Ham and Slaven Billic for sending the boys up to us at the start of the week because that's important for our preparation for the first game of the season," he added.
"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."
Newly-promoted Bolton begin the Championship season at home to Leeds United on Sunday.