The United States international becomes the MLS outfit's third designated player signing following the captures of Jermain Defoe and Gilberto.
Bradley's switch ends an 18-month spell with Roma and sees the 26-year-old return to the MLS after an eight-year absence.
"I have never been more excited, more determined or more motivated for any challenge in my entire career," he said.
"(I want) to be a part of this project and this city, to help bring a winning team."
"I told the players every day that we should always strive to perfection," Nelsen said.
"If I had the chance to pick three designated players, I think we would be getting pretty close to perfection with the players we've brought in."
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