While MLB.com has learned that no firm offer has been made, a baseball source confirmed the parties are in the process of exchanging figures.
According to reporter Eric Reynoso of Cafe Fuerte, the Marlins have a six-year, $40 million offer on the table for the 26-year-old. The Miami Herald reports the offer is less than $40 million.
For months, the Marlins have clearly stated that they are interested in signing Cespedes.
An offer was anticipated during his visit to Miami on Wednesday, where he toured the team's new ballpark and met with club officials. Until the Cafe Fuerte report, nothing was confirmed on a contract offer.
Cespedes on Thursday returned to the Dominican Republic, where he has established temporary residency.
"It was good to see him again," Marlins president David Samson said Wednesday on his weekly segment of The Dan LeBatard Show on 790 The Ticket. "We hadn't seen him since he was in the Dominican. He's still not a free agent, officially.
"But he was able to be in town for some appointments, I guess. He wanted to come and see the ballpark, and we're always happy to show people the ballpark."
Cespedes recently gained temporary residence in the Dominican Republic, and he received a travel visa, and arrived in Miami late Tuesday afternoon.
Cespedes and his agent, Adam Katz of Wasserman Media Group, spent Wednesday in Miami, getting a feel for the market and seeing the new ballpark.
Despite being declared a free agent by Major League Baseball on Jan. 25, Cespedes has not been legally cleared by the U.S. Treasury Department's Office of Foreign Assets Control.
Until Cespedes is unblocked, he can negotiate and even reach agreement with a team. However, he can't formally sign until he is declared eligible.
Joe Frisaro is a reporter for MLB.com. He writes a blog, called The Fish Pond. Follow him on Twitter @JoeFrisaro. MLB.com reporter Jesse Sanchez contributed to this report. This story was not subject to the approval of Major League Baseball or its clubs.