"Negotiations with Lufthansa AG and the British airline EasyJet will now continue until 12 October," the firm said in a statement following a board meeting.
Technical details around winding up the carrier to be thrashed out in the coming weeks and any final deal will need to be approved by European regulators.
EasyJet is interested only in parts of the passenger airline, while Lufthansa's bid includes subsidiary LGW and Austrian airline Niki as well as parts of Air Berlin.
At stake are the carrier's 140 leased aircraft -- including those wholly owned by Niki -- coveted landing and takeoff slots at German airports, some prime Berlin real estate and the livelihoods of thousands of employees.
"We are on the way to achieving good job prospects for around 80 percent of our colleagues with our bidders," Air Berlin chief executive Thomas Winkelmann said.
The company has some 8,600 employees, including part-time workers, according to DPA news agency.
Air Berlin suffered recently as huge numbers of pilots called in sick in a protest action sparked by the uncertainty over looming job losses.
At the time, Winkelmann said the wildcat strike posed an "existential threat" to Air Berlin.
"Stable flight operations in the coming days and weeks are essential for success" in the sell-off negotiations, administrator Frank Kebekus said Monday.
Air Berlin triggered bankruptcy proceedings in mid-August after losing a cash lifeline from its biggest shareholder Etihad Airways, giving potential buyers a month to make their offers.
In the race to strip down Air Berlin, Lufthansa reportedly beat out IAG -- owner of Iberia and British Airways -- and three bids of between 500 million and 600 million euros ($600 million and $715 million) apiece from private investors.
Despite media reports to the contrary, there was no sign Monday that German competitor Condor was still in the bidding process.
Irish low-cost airline Ryanair stayed out of the bidding as its outspoken chief Michael O'Leary denounced a German "stitch-up" designed to favour Lufthansa's expansion of its low-cost subsidiary Eurowings.
"Rumours that politics had influenced the selection of investors are… completely absurd," insolvency administrator Lucas Floether said in the statement.
The German government, which extended a 150-million-euro bridging loan to Air Berlin to keep its aircraft aloft, ruled out any one competitor taking over the carrier whole for competition reasons.
Neither details of the bids from Lufthansa and EasyJet nor the amounts of their offers have been made public.
But the airline said "there is a good chance" that the government's loan can be repaid to public investment bank KfW.
German newspaper Bild and Berlin tabloid B.Z. reported at the weekend that creditors expected the sale of Air Berlin assets to bring in 250-350 million euros.
Lufthansa chief Carsten Spohr told journalists last week that the carrier was interested in buying up to 78 Air Berlin planes.
Long-hauls to be scrapped
Lufthansa has signalled it could hire up to 3,000 people to go with its new planes, possibly including some from Air Berlin's ranks.
As European low-cost airlines engage in dogfights for control of the skies, trade union Verdi fears that some buyers plan to use their own staff to profit from Air Berlin's aircraft and landing slots, rather than offering existing employees opportunities.
Verdi's Christine Behle said last week the union expected the successful bidders to "take on responsibility for Air Berlin's employees and offer them good opportunities for the future".
But the clock is ticking.
Despite the government's cash infusion, Air Berlin is running on fumes. It has already been forced to slash most long-haul flights and plans to end the service all together by October 15.
Some domestic flights are also being cancelled from the end of the month.
Air Berlin, which transported 36 million passengers in 2016, has assured anxious travellers it will operate its other flights as scheduled until bankruptcy proceedings are completed.
Passengers who bought plane tickets after August 15 can be reimbursed for their reservations.