The deal includes a player option on the fourth year, a league source said.
Richardson's agents with BDA Sports, Bill Duffy and Erik Kabe, finalized several weeks of talks with Heat officials Wednesday.
Richardson has been one of the Heat's great scouting and player development success stories, rising from the 40th pick in the 2015 NBA draft out of Tennessee to one of the franchises' young cornerstones. He has won the trust of coach Erik Spoelstra in his development as a two-way player.
Richardson, who will turn 24 on Friday, averaged 10.2 points, 3.2 rebounds and 2.6 assists per game for the Heat last season. He started 34 games but played only 53 overall because of a recurring ankle injury.