The 24-year-old American has signed a three-and-a-half year contract at the Reebok Stadium following a successful trial period with the club earlier in the season.
Ream arrives at Bolton following the departure of key defender Gary Cahill, who completed a move to Chelsea earlier in the month.
"First and foremost we are absolutely delighted to secure Tim's services," Owen Coyle told the club's official website.
"He is a US international, a player that year-on-year has progressed and got better, and he will continue to do that.
"He was trialled by a number of clubs, including West Brom and also Arsenal who tried to sign him in the summer, so it is great that he sees his future here.
"Tim is everything that we want in the players we are bringing to the club when we spend in the transfer market. He is young, has a tremendous attitude and a winning mentality.
"I know he will fit in well with the rest of the squad here at Bolton."
Ream expressed his delight at joining Bolton, adding that he had been hoping for a move to the Premier League and felt "really comfortable" while on trial at the Reebok.
"I'd been waiting for a bit of paperwork to be completed and getting the work permit sorted was a huge relief," he said. "I'm just delighted to be here and I'm glad I have been given this opportunity.
"I knew there was interest in me when I came over last year and trained for a couple of weeks, and then after Christmas time things really did start to ramp up. Negotiations started and that's when I knew things were getting serious.
"From the first day here I felt really comfortable. The gaffer said that I had nothing to prove and it was a chance to get to know the guys and the coaching staff, and that put me at ease from the very beginning.
"Even before that week I aimed to challenge myself in England and if an opportunity arose I wanted to come over and try to make it here.
"Once that week was over I definitely had a better understanding of what it takes to play at this level."
The Missouri-born centre-back will not be ready for action immediately, with the MLS season having concluded in November, but hoped to get back to match fitness as soon as possible to help the Trotters steer clear of the relegation zone.
"I've always been in pretty good condition, even when I have taken a few weeks off - that's my forte. But I do have a couple of weeks to go until I am in tip-top game shape.
"The team is in a good run of form right now and I'm just looking to contribute in any way that I can."
Red Bulls general manager and sporting director Erik Soler admitted his disappointment at losing Ream but praised his "professional" attitude.
"We are extremely happy for Tim," Soler said. "He has worked very hard since he joined Major League Soccer and became a Red Bull and we firmly believe that this opportunity will be beneficial to him and the Red Bulls.
"This was a difficult decision for us as a club, but at the end of the day, we could not turn down this opportunity to help Tim’s development as a player.
"He has become a mature professional during his time here and we will fully support him as he moves to Bolton. He will always be welcome at Red Bull Arena."
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