The deal is for two years and $9.7 million, Bradley's agent, Bill Duffy of BDA Sports, told ESPN.
The deal includes a player option in the second year, Duffy said.
The Lakers used their room exception to sign Bradley.
The 6-foot-2 guard is known for his defensive versatility, having been reared in Doc Rivers' system. Noted offensive talent Kyrie Irving once said Bradley guards him better than anyone in the league.
He then started 14 games in Memphis, averaging 16 points, 4 assists and 3.2 rebounds a game in that span after overcoming knee and shin injuries.
The 28-year-old guard is a career 36.4% 3-point shooter. He gives Lakers coach Frank Vogel an additional switchable piece to use in a guards corps also featuring Kentavious Caldwell-Pope, Green, Quinn Cook, Rajon Rondo, Alex Caruso and Troy Daniels.
Bradley has 39 playoff games to his credit, averaging 12.2 points, 3.1 rebounds, 1.6 assists and 1.2 steals in the postseason. He started at guard for the Boston Celtics in their run to the Eastern Conference finals in 2017.
Bradley spent the first 6½ seasons of his career with the Celtics before being traded to the Detroit Pistons during the 2017-18 season. He was traded to the Clippers that same season in a deal highlighted by Blake Griffin's being sent to the Pistons.
The addition of Bradley brings the Lakers' roster to 14 players, with one open spot remaining.
ESPN's Dave McMenamin contributed to this report.