After denying service to a same-sex couple, the owners of Portland's Sweet Cakes are reportedly refusing a court order to pay the couple $135,000 in damages.

Owners Aaron and Melissa Klein earlier made headlines after refusing to bake a wedding cake for lesbian couple Laurel and Rachel Bowman-Cryer, citing their religious beliefs.

However, damages were awarded to the lesbian couple after state authorities found that they were discriminated against based on their sexual orientation, thus violating their civil rights.

Citing Oregon officials, Reuters said the Kleins filed an appeal to the ruling.

"It's difficult to understand the Kleins' unwillingness to pay the debt when they have, very publicly, raised nearly a half million dollars," said Labor Bureau spokesman Charlie Burr.

Burr was referring to the more than $500,000 that the Kleins reportedly had raised through crowd-funding.

"They are entitled to a full and fair review of the case, but do not have the right to disregard a legally binding order," Burr said.

Reuters reported that state officials are trying to place a property lien or attach other assets belonging to the Kleins in order to collect the funds.

Meanwhile, Herb Grey, the Kleins legal counsel, said the couple should not be obligated to pay the damages because the case is not settled.

He clarified that the Kleins had raised far less than $500,000.

"They continue to stand on their well-established constitutional rights to live and work based on their values and beliefs," Grey said. "Religion is a protected class just like sexual orientation is."

Grey said he expects the appeal to be heard in the spring.

The Kleins' case is one of many gay-related disputes nation-wide since the US Supreme Court's decision in June to legalise same-sex marriage in all 50 states.

Earlier, Kentucky clerk Kim Davis was jailed for six days after she refused to issue licences to gay couples.