Aaron Carter Denies Dog Abuse Allegations, Rescue Offers $10k For Pug

Aaron Carter is denying allegations he's abusing his dog, Nala ... and says he won't give the pug to a rescue org offering big bucks for the pup.

Here's the deal ... folks are accusing Aaron of abusing Nala by allegedly keeping her in a crate outdoors in 100-degree heat and locked in bathrooms for multiple hours a day. People are also alarmed over Aaron's new German Shepherds, worrying the big dogs could harm little Nala.

Some people even called Lancaster Animal Control on Aaron ... and Nala's got a trending hashtag, #SaveNala, which caught the attention of a pug rescue organization, The Pug Queen.

The org's founder, Izabella St. James, tells us she reached out to Aaron offering him $10,000 to rescue the pug and rehome Nala, but claims the singer threatened her with a lawsuit for defamation.

Aaron tells TMZ ... the abuse allegations are absolutely not true. He says he only had Nala in a crate for 15 minutes to separate her from his new guard dogs.

AC acknowledges animal control came to his house, but claims officers concluded he did nothing wrong ... and left after he explained he was simply separating his new pets.