Heat owner Micky Arison announced the news on Twitter
Several playoff teams pursued Johnson, but the Heat clearly moved to the front of his wish list.
Johnson cleared waivers Saturday at 5 p.m. ET after negotiating a contract buyout with the Brooklyn Nets. Sources told ESPN's Marc Stein that the Cleveland Cavaliers and Atlanta Hawks were the other finalists on Johnson's list, but all signs pointed to Johnson choosing Miami.
"To put somebody on the floor that can shoot the ball, can score in different areas of the floor and make plays, just adds to what we're trying to do," Dwyane Wade said after Miam's 101-89 loss in Boston. "Joe is a friend of mine. I tried to do my best to paint the picture that this is a good place to be, and the decision from there is his. Make sure he sees my name in his inbox a lot."
Johnson averaged 11.8 points, 3.9 rebounds and 4.1 assists for the Nets this season on 40.6 percent shooting overall and 37.1 percent from 3-point range.
Before negotiating a buyout with the Nets, the 34-year-old Johnson was making $24.9 million this season in the final year of a mammoth six-year, $124 million deal that was the league's largest at the time when he signed it in 2010.
Miami has two available roster spots after making trades last week that moved its payroll back under the league's punitive luxury tax line.
By shedding what essentially were the original salaries for Mario Chalmers and Chris Andersen, the Heat are in position to avoid the dreaded repeater tax status that would drastically hinder their ability to execute sign-and-trade deals and would also make it far more expensive to sign free agents.
Although there are complicated alternatives within the league's rules, signing Johnson prior to March 10 -- or adding any player who shakes free after being bought from a current team by Tuesday -- could throw the Heat back above the luxury tax line.
For that reason, Heat president Pat Riley acknowledged being in a bit of a dilemma when he met with reporters late Thursday. By waiting several days as part of a loophole to maintain future flexibility, the Heat would miss out on adding a player such as Johnson. By adding help now in a player such as Johnson, it would provide a major boost to an injury-ravaged roster that is fighting to remain in position for one of the top four playoff spots in the Eastern Conference.
The Heat still aren't certain if leading scorer Chris Bosh will return this season as he continues to seek treatment options for an undisclosed medical condition that has sidelined him since Feb. 9. Reserve guard Tyler Johnson underwent shoulder surgery earlier this season, and Miami announced Friday that backup point guard Beno Udrih had season-ending foot surgery.
ESPN's Marc Stein and Michael Wallace contributed to this report.