A Christian official has refused to issue marriage licences to same-sex couples in Kentucky despite exhausting all of her legal options.

Two gay couples denied licenses have requested Rowan County Clerk Kim Davis be held in contempt of court.

But they asked a district judge to punish Ms Davis by fining her, rather than sending her to jail.

The US Supreme Court on Monday rejected the argument that her faith prevented her from carrying out her duty.

Despite court orders to continue issuing licences, Ms Davis turned away two gay couples seeking marriage licences on Tuesday morning.

Ms Davis, who cannot be removed from post because she is an elected official, stayed inside her office with the blinds closed as the couples demanded to see her.

She eventually came out, but she said she would not issue any licences.

"Under whose authority?" she was asked. "Under God's authority," she said.

Dozens of reporters and activists, both supporters and opponents of gay marriage, filled the waiting area of the Rowan County clerk's office, alternatively chanting "do your job" and "stand firm".

One of the men seeking a licence, David Ermold, said he and his partner would not leave the office until they were married.

"Then you're going to have a long day," Ms Davis replied.

Since the court legalised gay marriage nationwide two months ago, Ms Davis has refused to issue any marriage licences to anyone.

The American Civil Liberties Union sued her on behalf of two gay couples and two heterosexual couples.

The clerk has been ordered to appear in federal court on Thursday to address a motion filed by one of the couples asking for her to be held in contempt of court.

In making its ruling on Monday, the Supreme Court denied her request for a stay while she pursues an appeal.