Asked whether he felt the sides were "far down the road" toward an agreement, Melvin said, "You never know. Sometimes things can happen in two hours, three hours, and some things can drag on.
"They know that they've had time. Things are coming into a little bit of a picture here, I think."
The Brewers are also looking at relief pitchers, including former Cubs closer Carlos Marmol, but the budget requires a resolution at first base before moving to other areas. Hart is clearly the club's top choice, but Melvin & Co. have also laid some groundwork for trades in the event Hart finds a richer deal elsewhere. The nature of the Brewers' offer was not immediately known.
Melvin and Hart's agent, Jeff Berry, have had at least two discussions this week.
"We'll probably get a sense of [Hart's decision] before we leave here," said Melvin, who will return to Milwaukee after Thursday morning's Rule 5 Draft. "We're hoping and thinking that will happen. … And they probably need to know, too."
Said Brewers manager Ron Roenicke: "We're eager to see what Corey wants to do."
Roenicke spoke personally with Hart several weeks ago.
"Corey knows where I stand with how I think about him," Roenicke said. "It always comes down to a decision on how much does he want to come back with us, and where, financially, we are with our offer. … Hey, he's a quality guy, and when he's healthy, he's a great player. There's some question marks, even though we know that the recovery [from knee surgery] is coming along really well."
Melvin has had only phone calls with representatives for other free-agent first basemen, so it appears the club would fill the position via trade if Hart does not re-sign. Melvin would not comment about the status of discussions with the Mets, who have talked with teams about Ike Davis and Lucas Duda, or the Marlins, who have Logan Morrison.
Whomever the club acquires, Melvin said, would have to be better than the Brewers' top in-house option, Juan Francisco. While prospect Hunter Morris is technically an option as well, the Brewers would prefer to send him back to Triple-A Nashville next season.
"Sometimes it takes time to fill a position when you've had a player like Prince [Fielder] in there a number of years," Melvin said. "When you're used to having someone at a position like that, you sort of take it for granted. It's like the Yankees and Jorge Posada, trying to replace a guy like Posada. I go back to the Orioles years ago and trying to replace Cal Ripken. We're trying to replace Prince Fielder with another Prince Fielder, and that's not going to [immediately] happen."
In the two seasons since Fielder left via free agency, 12 different players have started a game at first base for Milwaukee, including seven players in 2013 after Hart, Mat Gamel and Taylor Green were all injured.
Hart underwent microfracture surgery on his right knee in January and had a similar procedure on his left knee surgery in July. He missed the entire season.
Now the Brewers are looking for a short-term solution while Morris and Jason Rogers, the organization's last two Minor League players of the year, continue their development. In hindsight, Melvin said Tuesday, it might have benefitted Morris to join the Brewers as a September callup. At the time, the team was still evaluating Francisco.
"At some point they get an opportunity, and it's up to them to capitalize on that opportunity," Melvin said. "It's the same way that [outfielder] Khris Davis took 180 at-bats or whatever it was, and did something. Eventually, it gets to that point with any young player."
While waiting for word from Hart, Brewers officials spent time in the club's suite at the Walt Disney World Swan & Dolphin Resort brainstorming ideas and fielding occasional calls from other clubs. Melvin said he'd received some interest in third baseman Aramis Ramirez, who will be a free agent at the end of 2014, but would have a hard time trading Ramirez because the Brewers don't have anyone to replace him.
Another infielder entering a contract year, Rickie Weeks, has not drawn much interest. He had surgery in August for a torn hamstring. The Brewers are preparing to begin next season with Scooter Gennett at second base, leaving Weeks' future in limbo.
"It's hard when he didn't play the last couple of months," Melvin said. "I trust that he'll be OK. Mark [Attanasio, Milwaukee's principal owner] and Ron [Roenicke] saw him at Ryan [Braun's] wedding and said he looked very good, like his weight was down. I don't know how they could tell if he was moving well or not -- the dance floor?"
Asked whether he was getting any interest on pitchers Yovani Gallardo (signed through 2014 with a 2015 club option) or Kyle Lohse (signed through 2015), Melvin said, "I don't want to get into that. We're not motivated to move them."
At the moment, the Brewers plan to use Lohse, Gallardo, Wily Peralta and Marco Estrada in the rotation, with Tyler Thornburg getting first dibs on the fifth spot and Will Smith and Johnny Hellweg in reserve.
The bullpen is less settled, with Marmol (who is represented by Paul Kinzer, Ramirez's agent) among the free agents with closing experience under consideration. The Brewers consider other free agents like Grant Balfour, Fernando Rodney and Joaquin Benoit out of their price range.