"A substantive accord is taking shape with the European Commission," Lufthansa said in a short statement, adding that accordingly it had asked Austria for a July 31 deal deadline to be extended to the end of August.
The European Commission is probing the proposed link-up amid fears the Lufthansa takeover could lead to higher prices for passengers or fewer flights on some routes.
It has demanded that Lufthansa make concessions, with those offered to date deemed insufficient by the commission.
A spokesman for the commission in Brussels said Tuesday "a new offer" had been received from Lufthansa that was "significantly different" from what had been submitted previously.
On Monday, reports in Austria said Lufthansa was prepared to give up key slots in order to get the AUA deal approved in Brussels.
The German carrier was ready to hand six of its 10 take-off and landing slots between Vienna and Frankfurt to its competitors, including Austrian low-cost airline Niki and Slovenia's Adria Airways, one report said.
Austrian Finance Minister Josef Proell told national television ORF on Monday that the two parties were "closer than at any point in the last few weeks," adding: "We are in the home stretch."
The Austria state owns a 41.6-percent stake in AUA through its state holding company OeIAG.