The Marlins presented Reyes with preliminary contract numbers to chew on.
The figures, according to Ken Rosenthal and Jon Paul Morosi of FoxSports.com, are six years and $90 million.
If accepted, the offer would make Reyes the highest-paid player in Marlins history.
Since Reyes visited Miami on Wednesday, the Marlins have been very confident of their chances to sign the All-Star shortstop, who has spent his entire career to this point with the Mets.
The FoxSports.com story also added that Reyes is intrigued about playing in warm weather, which would help his hamstring problems, and the possibility of playing for manager Ozzie Guillen.
Now the Miami Marlins, the organization is rebranding itself, and with its new retractable-roof stadium, the team feels it has plenty of selling points for free agents.
Reyes took a tour of the new ballpark during his one-day visit, and he ate lunch at world-famous Joe's Stone Crab.
Along with Reyes, the Marlins also have preliminary contract offers out to left-hander Mark Buehrle, first baseman Albert Pujols and right-handed reliever Ryan Madson.
Reyes, Buehrle and Pujols visited Miami and toured the ballpark last week, but Madson didn't travel to South Florida.
Since the start of free agency, the Marlins have been arguably the most aggressive team on the market, although they have yet to sign a player.
Along with hosting Buehrle, Reyes and Pujols, team officials also traveled to the Dominican Republic last week to watch a private workout of center fielder Yoenis Cespedes, a Cuban defector who has yet to be cleared as a free agent.
The Marlins are not commenting officially on whether they have made offers or not. But the team noted that its new ballpark is giving it revenue streams to go after high-profile players.
"Our payroll is in a position where our job is to build a team with contingency plans," team president David Samson said on Friday at the renaming ceremony. "So we're going out and being very aggressive with free agents who we think can help the team. We have a first choice. We have a 1A and a 1B."
Chances are, the Marlins won't be able to sign all four players to whom they have presented offers. But they could add two, and perhaps a third in Madson as a closer.
Reyes has emerged as the top choice.
If they land the 28-year-old Reyes, the Marlins would be bringing in the 2011 National League batting champion. He hit .337 this past season with the Mets.
The addition of Reyes, however, would create an internal issue.
The Marlins already have a three-time NL All-Star shortstop in Hanley Ramirez.
Ramirez would likely be asked to move to third base.
A shortstop all his life, Ramirez told reporters last week: "I'm a shortstop right now, so I don't know. I consider myself a shortstop."
Having two star players both wanting to be at the same position could present some problems. But the Marlins feel that they have the perfect manager in Guillen to make both players accept their roles.
The Marlins will have competition for Reyes, as the Brewers have made preliminary contact with the shortstop's agent, Peter Greenberg.
But no offer was made.
The Marlins, meanwhile, are moving aggressively, continuing to sell free agents on the appeal of Miami, the new ballpark, warm weather, Guillen as a manager and the possibility of turning the franchise around in a hurry.
"What we have done is take the time in an organized way," Samson said a few days ago. "We knew ... going into the offseason what we were going to do."
Now, it's a matter of closing some deals.