Athletes First spokeswoman Savannah Wall confirmed the agreement on Monday after Moore wasn't picked in last week's NFL draft. Moore is represented by agent David Dunn, chairman of Athletes First.
Moore won 50 games in college, a Football Bowl Subdivision record for a starting quarterback, but wasn't among the 253 players taken in the draft.
"Very frustrating process, but a great outcome in the end," Moore told the Idaho Statesman. "Going into it, you felt very comfortable and confident that you'd get an opportunity to get drafted. Bottom line is it didn't happen. But it's kind of funny you end up where you'd love to go in the first place."
The Lions acknowledged Saturday night that adding a No. 3 quarterback was on their list.
"That will be one of the things that we'll probably be looking for," Detroit coach Jim Schwartz said.
Moore, who completed nearly 74 percent of his passes last year, finished his college career among the NCAA all-time leaders with 14,667 career yards passing and 142 career touchdown passes. But the 6-foot, 197-pound native of Prosser, Wash., knew many questioned his future in the NFL because of his height and a perceived lack of arm strength.
"There's a lot of ways to evaluate a quarterback," Moore said at the NFL combine in February. "Some evaluate it and put a lot of emphasis on that. That's fine. Smile, and move on."
He told the Statesman this weekend: "I don't think there will probably be a more motivated quarterback."
Boise State had a school-record six players picked in this year's draft. Detroit selected former Boise State receiver Titus Young in the second round last year and he had a productive rookie season as a No. 3 receiver behind All-Pro Calvin Johnson and solid veteran Nate Burleson.