The sources told Reuters on Friday the consortium, named for the former Hapag boss credited with having invented the cruise, may buy 20 percent of the world's fifth-biggest container shipper, less than expected.
TUI, which holds 38.4 percent of Hapag-Lloyd, has said it wanted to divest a 33 percent stake.
"We are in advanced talks," said a spokesman for the city of Hamburg's finance authority.
A spokesman for TUI AG said talks were going well.
TUI, which aims to focus on its tourism business, originally tried to sell a majority stake in Hapag-Lloyd in 2008. That attempt was derailed by the global financial crisis, meaning TUI kept a larger stake than it originally intended.
For months, it has been inching toward a deal with the Albert Ballin consortium, a group of investors that also includes Klaus-Michael Kuehne, the majority owner of Swiss logistics group Kuehne & Nagel .
To win more time to negotiate, the parties are pushing back by a month a deadline for an agreement on the price of any deal.
Originally, they said an independent auditor would be brought in to determine the market value of TUI's stake if they failed to agree on a price by the end of this month.
"I am confident that it will not be necessary to call upon the auditor," Kuehne & Nagel chairman Karl Gernandt, a confidante of Michael Kuehne, told Reuters.
TUI chief executive Michael Frenzel has been under pressure from shareholders to complete the years-long process of offloading Hapag-Lloyd but public finances are weighed down by weaker economic growth amid the euro zone debt crisis, and talks with investors based in China and Oman did not result in a deal.
TUI, which controls London-listed tour operator TUI Travel , will hold its annual shareholder meeting on February 15.
TUI Travel, which takes 30 million people on holiday each year, reported a better than expected full-year profit in December, in stark contrast to rival Thomas Cook.