Ms. Weinstock said she hoped the $5 million adoption center the city is planning to build beside the Manhattan shelter would have its entrance on 109th rather than 110th Street, to make it seem more separate from the shelter itself.

“We want the public to see us the way they see Animal Haven and A.S.P.C.A. — warm and friendly,” she said.

Warm and friendly can be tough to pull off when the agency must take every animal that is brought in — something Animal Care Centers’ contract with the city requires — and it does not have the room or the capability to treat them all.

“Caring for 35,000 animals is an impossible task,” Ms. Weinstock said.

Last year, the shelters euthanized 15 percent of cats and 9 percent of dogs. (The shelters also take in hundreds of rabbits but do not euthanize any unless it is medically necessary.)

The daily “at-risk” lists that the agency puts out — candidates for euthanasia if not adopted by the next morning — often include animals who were listed as “normal” on an initial medical exam and as having “major conditions” days later. Upper respiratory infections like kennel cough spread quickly through the shelters and can easily turn into pneumonia.

“All the brand-new dogs and the sick dogs are traveling in the same hallway all the time,” said a volunteer at the shelter in East New York, Brooklyn, who declined to give her name for fear of losing her post. “If a dog has kennel cough and they’re in the adoptions room, sometimes it’s not recognized for a day or so.”