Non-tendered by the Marlins last Friday, Nelson's agent, Doug Schaer, has fielded inquiries from 19 teams.
The 34-year-old reliever has not ruled out returning to Florida.
"We have yet to solicit any offers," Schaer said. "Still, we've been receiving offers."
Without giving specifics, Schaer said a few teams have presented seven-figure deals.
In the mix are the Royals, Cardinals, Red Sox, Braves, Rays and Mets.
Schaer is hopeful Nelson will sign before Christmas.
"We're on a five-to-seven-day path," Schaer said.
Nelson enjoyed a strong 2008 with Florida. In a setup role, the right-hander was 3-1 with a 2.00 ERA in 59 games. He also collected one save.
He's had experience closing, which is attractive to teams interested in his services.
As part of his decision, Nelson also is trying to position himself for after his playing career. When he retires, he hopes to become an executive with a club. He would be interested in being a special assistant or Minor League coordinator.
The Marlins initially offered Nelson $650,000, but raised their figure to $800,000. By midnight Friday, however, Florida declined to tender Nelson a contract.
Joe Frisaro is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.