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Harvey and Bob Weinstein are a couple of steps closer to winning back the company they named after their parents, Miriam and Max.

Negotiations for Miramax Films continued to take place over the weekend between Disney, the Weinstein brothers, supermarket magnate Ron Burkle and his Yucaipa Cos., New York investment firm Fortress Investment Group and Colbeck Capital Management.

The Mouse House, late last week, granted the Weinsteins an exclusive five-day window to hammer out terms and wrap up a deal to unload the specialty label as early as this week.

The Weinsteins and their financial backers have sweetened their initial $600 million offer several times, first converting it from payouts over time to upfront cash and more recently upping it to $625 million.

Through the latest round of dealmaking, Disney is looking to hold onto some of the remaining titles that Miramax has yet to release, including the Elton John-produced toon “Gnomeo and Juliet,” the John Madden-helmed thriller “The Debt,” starring Sam Worthington and Helen Mirren, comedy “The Switch” (formerly “The Baster”) starring Jennifer Aniston and Jason Bateman, thriller “Don’t Be Afraid of the Dark,” “Tell Me” (formerly “Last Night”), and Julie Taymor’s “The Tempest.” For others, Disney would receive a 10% distribution fee.

It also was negotiating for several of the properties it co-owns with the Weinstein brothers. The total Miramax library includes 611 movies, such as “Chicago,” “Pulp Fiction,” “The English Patient,” “Good Will Hunting,” “Gangs of New York,” “Swingers,” “Shakespeare in Love,” “The Crying Game,” “The Talented Mr. Ripley” and “Cold Mountain. ” It includes a large number of direct-to-DVD titles, episodes of TV shows like “Project Runway,” produced by Miramax Television and books published through Talk Miramax Books. There also are plays that Miramax co-produced, like Tom Stoppard’s “The Real Thing,” when it got into the live theater business in 1999.

But Disney could end talks at any time without a deal and seriously consider its two other offers.

Those include a joint $550 million cash offer from Alec and Tom Gores through their Gores Group and Platinum Equity investment arms, and $650 million from a consortium of financial partners assembled by David Bergstein.

The Weinsteins, however, were always expected to have the upper hand in the talks — as long as they could come up with a favorable price.

While Disney has been looking to unload Miramax on the highest bidder, it also has a long relationship with the Weinstein brothers.

The brothers co-founded Miramax in 1979 and sold it to Disney in 1993 for $80 million. While the Weinsteins walked away from the company in 2005 to form the Weinstein Co., they have the right to reboot or produce sequels based on 25 films in the Miramax library, including “Scream,” “Scanners,” “Piranha,” “Scary Movie,” “Halloween” and “Spy Kids.” They can also acquire new projects from 76 pics that Miramax had in development until July 1, sources said.

Sources also cautioned that the deal under consideration is highly complex and that there’s no assurance that the brothers and their backers will come to terms on an agreement.

All parties declined to talk about the latest details of the dealmaking because the talks are private.