The center fielder from Cuba told a reporter from The Associated Press in the Dominican Republic that the Marlins, Cubs, White Sox, Orioles, Tigers and Indians have expressed the most interest.
Cespedes, 26, made his debut on Thursday for Aguilas Cibaenas in the Dominican Republic Winter League.
"I decided to help the Aguilas [Cibao Eagles], and I don't care if my value goes down," Cespedes said. I'm going to try to give the best of myself. The Aguilas opened the doors for me and I want to help them."
Cespedes went 0-for-3 with three strikeouts in his debut.
"It had been 10 months since I'd played a night game," Cespedes said, "so my sight wasn't very good, but things will improve now."
The outfielder is in the process of obtaining residency in the Dominican, which could come as early as the next few days. Once residency is established, he will petition Major League Baseball for free-agency status.
With Miami's strong Latin influence, the Marlins have made it clear that they are heavily interested in Cespedes. Team president David Samson said during his segment this week on "The Dan LeBatard Show" on 790 The Ticket that the club will pursue Cespedes once he becomes a free agent.
"Aggressive right to the point of stupidity, but not quite there," Samson said of the team's expected efforts. "We think he's a perfect fit for us, but it has to be sane. [We've] expressing interest, going to visit, making it very clear to his representatives and to him and his family that we think he should not be anywhere other than Miami.
"As a Cuban and someone in the D.R., it makes perfect sense. We have a perfect position for him to play. It would be great."
Marlins officials, including owner Jeffrey Loria and manager Ozzie Guillen, attended a private workout for Cespedes in the Dominican in November. The club has maintained serious interest since.
Samson likes Miami's chances of signing the outfielder.
"I think that we're right in there," Samson said. "There are a lot of teams who want him, and we're definitely competitive and we're not fooling around. We expect he will be a significant sign and receive a significant signing bonus and contract, and we're prepared for that.
"Again, if you're dealing with another team who needs to make a bigger splash or wants to get irrational, we're not going to match that."
Cespedes is expected to sign a deal that will surpass the $30 million fellow Cuban Aroldis Chapman received from the Reds in 2010.