Copenhagen Zoo says it "had to euthanise" two cubs and their parents after it failed to re-home them.
The 16-year-old male and 14-year-old female were nearing the end of their natural lives in captivity, it added.
Last month, the zoo killed a healthy giraffe because it was deemed surplus to requirements.
"Because of the pride of lions' natural structure and behaviour, the zoo has had to euthanise the two old lions and two young lions who were not old enough to fend for themselves," the zoo said in a statement.
According to zookeepers, the male cub "would have been killed by the new male lion as soon as he got the chance."
The zoo said it had asked other parks to take the 10-month-old cubs, but had received no offers.
The new male lion is due to arrive in the next few days and will be introduced to the zoo's two female lions who, born in 2012, have reached breeding age.
Last month, the manager of the zoo received death threats after a healthy two-year-old giraffe called Marius was put down and later dissected in front of visitors.
The zoo said the animal did not fit in with the park's breeding programme but the move shocked animal lovers across the world.
Despite the outcry over the death, the zoo said it did not believe it was necessary to change its procedures.
A spokesman for the zoo said there would be no public dissection of the lions since "not all our animals are dissected in front of an audience".
A zoo in Denmark that provoked outrage after putting down a healthy giraffe has killed a family of four lions to make way for a new young male lion.