Wade could look to schedule meetings next week after he returns from vacationing in Europe, sources said.
On Friday, the Heat reached a verbal commitment with center Hassan Whiteside on a four-year contract that could be worth up to $98 million. The decision likely will force the Heat to renounce Wade's rights regardless of Durant's decision.
After the Whiteside contract, the Heat have $19 million in cap room, although they have some options to create more space if needed.
According to sources with knowledge of the situation, Wade does not want to accept less than the $20 million he made in Miami this past season.
Wade, who has played 13 years in Miami and won three championships, has grown frustrated with the Heat, as they prioritized Durant and Whiteside in free agency, sources said. Over the years, Wade has twice taken pay cuts to help the Heat manage their salary cap, and he has taken short-term deals to help the team with flexibility.
He has signed one-year contracts in each of the past two summers. He averaged 19 points, 4.1 rebounds and 4.6 assists playing in 74 games, his most in five years, this past season.
The Knicks are in the market for a shooting guard after trading for Derrick Rose and lining up a deal with Joakim Noah.
Milwaukee, which is close to his native Chicago, has special interest for Wade because of his college experience; he starred at Marquette. The Bucks have been active in the free-agent market, verbally agreeing to deals with Mirza Teletovic and Matthew Dellavedova on Friday.
Wade's representatives put out feelers to several teams right before the start of free agency, including one to the San Antonio Spurs. The Spurs are not interested, according to multiple NBA sources.
ESPN Staff Writer Michael C. Wright contributed to this report.