Hart, who contributed 30 home runs and 83 RBIs while transitioning from right field to first base in 2012, is one of three Brewers free agents, along with reliever Michael Gonzalez and infielder Yuniesky Betancourt.
"I talked to Corey's agent [Jeff Berry] last week and just said we would check in again a little bit further down the road," Melvin said. "I've seen some video on him. He's working out; it looks like he's moving around. We'll eventually follow up on it."
During a September visit to Miller Park, Hart suggested he was "optimistic" about re-signing with Milwaukee and said he would be "generous," but stopped short of providing financial parameters.
"I would take a discount to stay here, because I think I owe it to them to stay here and be a cheaper player, because -- nobody wants to play for free -- but I've basically sat there and watched all season," Hart said then. "I think I owe it to them and the fans to come back. That's kind of what we're hoping for, but at the same time, you don't know what's going to happen."
At the time, Hart was just shy of getting clearance to begin running. He underwent right knee surgery in January to repair a depression on the joint surface, then needed a similar procedure on the left knee in July for an injury that developed during rehab.
First base is wide open for the Brewers, who used a patchwork group of players at the position in 2013 that included shortstops Betancourt and Alex Gonzalez, third baseman Juan Francisco and catcher Jonathan Lucroy. They combined for a .629 OPS, lowest of the 30 Major League teams.
Francisco is arbitration-eligible as a Super Two player and batted .309 with two home runs and 12 RBIs in his first 14 games in the Dominican Winter League while implementing some adjustments to his batting stance prescribed by Brewers hitting coach Johnny Narron. Brewers coach Mike Guerrero is managing Licey, Francisco's team. Another internal option is former Minor League Player of the Year Hunter Morris, but his OPS slipped more than 150 points with a promotion to Triple-A Nashville last season, and was not among Milwaukee's September callups.
If the Brewers opt to fill first base from the list of free agents, "It's not a very deep list," Melvin said. "You would think that there would be more first basemen, but [Justin] Morneau is out there, [James] Loney is out there, and it pretty well stops at that point, I think."
He added: "The whole first-base situation, we have to take a look at."
As of 11 p.m. CT on Monday, Hart and all of baseball's other free agents were free to sign with any team. It was Hart's first foray into free agency after 14 professional seasons with the Brewers beginning in 2000, when he was Milwaukee's 11th-round Draft pick.
In his six full seasons with Milwaukee, Hart averaged 24 home runs and 78 RBIs with an .830 OPS. He will turn 32 on March 24, about a week before Opening Day.
Left fielder Ryan Braun's suspension quietly ended on Oct. 31 when he was reinstated from the restricted list and returned to Milwaukee's 40-man roster along with Hart and second baseman Rickie Weeks, who came off the 60-day disabled list. Braun was banned from the Brewers' final 65 games for violations of the Joint Drug Prevention and Treatment Program and baseball's Basic Agreement.