Verdi says the union will now hold a strike vote, which could lead to a work stoppage affecting air traffic during the busy summer travel period.
Lufthansa's latest offer was a 6.7 percent pay increase for around 50,000 ground and cabin personnel, a 1.2 percent increase over its previous offer. The increase would take place in two installments over 21 months.
Verdi, on the other hand, is demanding a wage increase of just under 10 percent for a period of 12 months.
Lufthansa's personnel director, Stefan Lauer, called the possible strike vote "completely excessive," saying the union should consider whether it wants its actions to hit vacationers in the middle of the peak summer holiday season.