Martin has spoken with executives from each of the clubs and is hoping to make his decision next week.
The veteran of 11 NBA seasons will be eligible to sign and begin playing with a team as soon as the Xinjiang Guanghui Flying Tigers complete their season in the Chinese Basketball Association.
Martin signed a $2.6 million deal with Xinjiang during the lockout, but left the team in late December in hopes of returning to the NBA.
The CBA's rules, however, prohibit him from joining an NBA team until Xinjiang's season ends, whether after the regular season or the playoffs. Xinjiang is currently out of the playoffs, which means Martin could potentially sign with an NBA club when the CBA season ends Feb. 16.
Martin, 34, is coming off the least productive season of his career, having averaged just 8.6 points and 6.2 rebounds in Denver last year. But contenders view him as a strong and tough defender who plays unselfishly and has a winning pedigree -- Martin twice reached the NBA Finals with the New Jersey Nets.
Martin's priority is to join a contender on which he'll have a sizeable role, according to a person close to Martin.
The Clippers and the Knicks can offer the $2.5 million mini-midlevel exception. San Antonio can as well, but it is unlikely to do so because of the luxury tax, sources said.
Atlanta is also limited by luxury tax concerns. Miami can only offer the veteran's minimum of $1.4 million.
Chris Broussard is a senior NBA writer for ESPN The Magazine.