“We have had rather too many incidents. But it is rather difficult, because it is dangerous to fire (a weapon) next to houses and people,” Stefan Holm, head of the police in Lerum, told regional newspaper GT.
A dog walker was attacked by a member of the elk family late on Friday. The man was injured and had to be taken to hospital. The dog died. Another dog was killed in an elk attack earlier this autumn.
The family, a female and her two youngsters, have been in the area since October.
Police hunters now have a week to shoot the animals before the order expires.
It is not uncommon for elk to behave aggressively towards humans, although injuries are rare.
But in 2009 a 69-year-old man was held on suspicion of murdering his wife in south-eastern Sweden, only to be exonerated after a forensic analysis revealed she had in fact been killed by an elk.