It is unclear what the monetary figure of the offer is, and a key question is whether the Marlins would agree to include a no-trade clause in their offer. The club did not offer shortstop Jose Reyes such a clause, which Pujols had with the St. Louis Cardinals.
The Marlins' latest offer is an increase from the nine-year proposal made in their first meeting with Lozano.
Sources told ESPN.com's Jayson Stark the Marlins expect a resolution to their bid to sign Pujols, one way or the other, late Tuesday night.
Meanwhile, the Chicago Cubs spoke to Lozano on Monday and made a qualifying offer to Pujols, according to reports.
Clubs that have spoken to the Marlins at the winter meetings have come away with the clear impression they are pressing to sign Pujols quickly. In a public appearance following the team's official announcement of the signing of new closer Heath Bell on Monday, Marlins owner Jeffrey Loria did nothing to discourage talk that they have payroll room to sign Pujols, even after reeling in Bell and Reyes; the latter agreed Sunday to a six-year, $106 million contract.
The signings of All-Stars Reyes and Bell come as the Marlins prepare to open the 2012 season in a new downtown Miami ballpark.
Baseball's new Big Fish were the talk of the winter meetings Tuesday, with teams wondering how close the Marlins were to an agreement with Pujols.
St. Louis said it submitted a new offer Tuesday to keep Pujols with the Cardinals, a team he's already led to two World Series titles in the last six seasons.
"They have a new stadium. They're excited about it, and it's good for baseball," former Marlins and current Yankees manager Joe Girardi said. "They had hoped that they'd get the new stadium and they would be able to do those types of things. Our roster is pretty set. We have a lot of guys that are on long-term deals. That's why maybe there's not a lot happening for us."
Having already reached deals with Bell and Reyes, the Marlins' interest in Pujols is real. What's unclear is whether the first baseman is prepared to go to Miami or whether his talks with the Marlins are an attempt to push the Cardinals higher.
"I know the ownership group is putting their best foot forward and trying to do everything that they can to make this possible," new Cardinals manager Mike Matheny said, "but I also know it's a complicated decision on both sides. There's a lot going on, a lot that I'm not even involved in, but I think it's clear to say that St. Louis Cardinals would love to have Albert, and we'll see how it all plays out."
Prince Fielder was still in play in the hitters' market, and C.J. Wilson and Mark Buehrle were among the available starting pitchers on the second day of the four-day swap session, which has been relatively slow.
When the meetings reach their last full day Wednesday, roughly two dozen free agents must decide by midnight EST on salary arbitration offers from their former clubs. David Ortiz, who has said he wants to stay with the Boston Red Sox, appeared likely to accept.
Information from ESPN The Magazine's Buster Olney, ESPN.com's Jayson Stark and The Associated Press contributed to this report.