The deal is worth $3.8 million, league sources told ESPN NFL Insider Adam Schefter.
The deal, struck Wednesday, will become complete if Bulger passes a physical.
Bulger has 27 games of at least 300 yards passing during his career. In 2008, he became the fifth-fastest quarterback in NFL history to reach the 20,000-yard passing milestone, doing so in only 81 games.
He fought a series of injuries last season and played in only nine games before placing in injured reserve on Dec. 26.
Information from The Associated Press was used in this report.