A deadline to submit bids to acquire Air Berlin or parts of it has been set for 15 September.
The airline said around 200 pilots, mainly captains, had called in sick.
Lufthansa's low-priced subsidiary Eurowings, which is renting aircraft as well as pilot and crew from Air Berlin, said it had also been forced to cancel some flights.
Air Berlin announced on Monday that it would end its flights to the Caribbean from Düsseldorf beginning September 24.
"This day is costing us several million euros", Winkelmann said in a statement.
"A stable operation is an essential precondition for successful negotiations", Mr Winkelmann said.
"VC is convinced that talks about a social plan for an orderly staff transfer is the only way to safeguard as many jobs as possible", says the union, adding that it has urged its members at Air Berlin to honour obligations in their labour contracts unless there is an "acute reason" to call in sick.
The airline had long struggled for survival, and booked losses amounting to 1.2 billion euros ($1.4 billion) over the past two years.
This has the potential to unsettle any potential rescuers for Air Berlin, which filed for administration last month when it was given a €150m (£135m) loan by the German government.
German flagship carrier Lufthansa - which already leases 38 of Air Berlin's 140 planes - could buy up to 70 aircraft with as many as 3,000 crew for Eurowings, German media had reported.
"We worked hard to be able to make this offer, with help not only from experts but also supportive comments and suggestions from employees, passengers and business partners of Air Berlin", the Bavarian airline tycoon wrote in a Facebook post late Sunday.