MLB.com has learned the Marlins and Giants are actively talking with the D-backs about the 28-year-old right-hander, whose fastball was recently clocked at 97 mph in a Spring Training game.
There has been speculation that the Red Sox also covet Julio, who would become the front-runner to be Florida's closer. However, a league source said Thursday that Boston isn't in the mix for Julio. The source added that the Red Sox are finding themselves mentioned in numerous trade rumors for the purpose of driving up the asking price.
A sticking point the Marlins appear to be facing is how much of the $3.6 million on Julio's contract the club has to absorb. It is believed that the Giants would take on more of the money.
The Marlins, though, may be able to pull off the deal if they are willing to part with right-hander Yusmeiro Petit.
Julio had 16 saves in 20 chances for Arizona in 2006.
Florida is actively trying to gain more experience in the back end of its bullpen. While Matt Lindstrom and Henry Owens have strong arms, both are rookies.
Along with Julio, the Marlins have been exploring trade options with the Giants for Armando Benitez.
Benitez was the Marlins' closer in 2004, setting a franchise record with 47 saves. But he's been injured in the past few years and there are concerns if he would hold up.
Florida's closer options right now are Lindstrom, Owens, Kevin Gregg, Randy Messenger and Taylor Tankersley.
Tankersley, a lefty, is likely going to be used in a late-innings setup role.
Tankersley also hasn't pitched in a spring game due to shoulder inflammation. He is scheduled to long-toss from 120 feet Friday, and if that goes well, he will pitch off the mound Saturday.
It is still questionable if he will be ready by the April 2 season opener at Washington.
The Marlins have had strong interest in Julio for at least a week.
Joe Frisaro is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.