The fourth year of the deal is a player option, league sources told ESPN.
Bazemore declined four-year, $72 million offers from the Rockets and Lakers, a source told ESPN, confirming an NBA.com report.
Bazemore played 23 games for the Lakers after they traded for him in 2014, and a source said he "seriously considered" their offer before deciding to return to the Hawks.
Combined with their agreement with Dwight Howard on a three-year, $70.5 million deal earlier Friday, the Hawks have nearly exhausted their available salary-cap space, while All-Star free agent and nine-year team veteran Al Horford has yet to pick a team.
However, the Hawks were engaged in trade talks Friday that potentially would allow them to open more cap space in the event Horford decided to re-sign. Horford is one of the most coveted free agents on the market, with several teams interested in him, including the Boston Celtics, Oklahoma City Thunder, Detroit Pistons and Rockets.
Horford met with the Rockets on Friday and expects to make a decision this weekend, a source said.
Bazemore averaged 11.6 points and 5.1 rebounds per game for the Hawks last season.
ESPN's Calvin Watkins, Ramona Shelburne and Jeff Goodman contributed to this report.