Shams Charania of The Athletic is reporting Butler has signed a four-year deal with the Heat.
Butler helped Philadelphia reach the Eastern Conference semifinals this season before the 76ers eventually lost to the Toronto Raptors.
Miami was seeking a replacement for the now retired Dwyane Wade and appears to have that in Butler.
The 29-year-old Butler averaged 18.2 points, 5.3 rebounds and 4.0 assists in 55 regular season games last year for the 76ers.
Richardson, a 25-year-old shooting guard, just finished his fourth NBA season with Miami and averaged 16.6 points, 3.6 rebounds and 4.1 assists in 73 games.