Free-agent signings can't commence until Thursday in the NBA, but the two players told the network they want to be teammates for the Heat.
Miami President Pat Riley issued a statement saying the Heat "are engaged in free-agent negotiations" with the pair.
"We intend to enter into player contracts with Dwyane and Chris once the moratorium period has ended," Riley said.
Wade, a six-time all-star, has averaged 25.3 points and 6.6 assists in seven years with the Heat.
Bosh, a four-time all-star, has averaged 20.2 points and 9.4 rebounds in his seven years with Toronto.
Miami, looking to regain its 2006 NBA title form, went 47-35 last season and were bumped out of the playoffs in the first round.