Bray, 28, has been with the Bears for the past two seasons. He hasn't appeared in a regular-season game with Chicago.
Bray entered the NFL in 2013, spending five seasons with the Chiefs before joining the Bears in 2018. He has played his entire NFL career alongside Bears coach Matt Nagy.
Bray will be a part of a quarterbacks room that includes Mitchell Trubisky and the recently acquired Nick Foles.
The Bears haven't yet made the signing of Bray official.