Texas Republican representative Joe Barton reportedly told a woman he would complain to the US Capitol Police if sexually explicit photographs of him and other material from their relationship was exposed publicly.

The Washington Post report of the threat came after Barton apologised for a nude photo of him that was posted on social media on Tuesday (21 November).

The photo appeared on an anonymous Twitter account and it was not immediately known who posted it or when it was taken.

Barton issued a statement saying that while separated from his second wife, prior to their divorce in 2015, he had sexual relationships "with other mature adult women". The 68-year-old Republican said each relationship was consensual and has since ended.

"I am sorry I did not use better judgment during those days. I am sorry that I let my constituents down," said Barton, the longest-serving member of Congress from Texas.

A woman, whom the Washington Post did not identify, has since told the newspaper of Barton's threat over the sexually explicit photographs, videos and messages he had sent to her.

She described sexual encounters and contact with Barton extending over five years, beginning with her posting of a message on Barton's Facebook page in 2011.

She also shared with the Washington Post a secretly recorded telephone conversation with Barton in 2015 in which he warned her against using the sexually explicit material "in a way that would negatively affect my career".

"I want your word that this ends," he said, according to the report of the recording, adding: "I will be completely straight with you. I am ready if I have to, I don't want to, but I should take all this crap to the Capitol Hill Police and have them launch an investigation. And if I do that, that hurts me potentially big time."

"Why would you even say that to me?" the woman reportedly responded. "The Capitol Hill police? And what would you tell them, sir?"

Barton replied: "I would tell them that I had a three-year undercover relationship with you over the Internet that was heavily sexual and that I had met you twice while married and had sex with you on two different occasions and that I exchanged inappropriate photographs and videos with you that I wouldn't like to be seen made public, that you still apparently had all of those and were in position to use them in a way that would negatively affect my career. That's the truth."

The woman, who is not married, reportedly spoke on the condition of anonymity to protect her privacy. She said she never had any intention of making the material sent by Barton public.

In a statement, Barton said a transcript of the telephone conversation provided by the newspaper may be "evidence" of a "potential crime against me". He also said that Capitol Police had informed him Wednesday that they were opening an inquiry.

The Twitter photo of Barton could violate a 2015 Texas law banning so-called "revenge porn", an offence that sees sexually intimate images or videos made public without consent.

Earlier in the day, Barton's spokeswoman told The Dallas Morning News that the congressman has no plans to step down.

Barton, a member of the conservative House Freedom Caucus, announced his re-election bid this month.

He was a consultant in the oil and gas industry before he joined the US House in 1985, and is currently the vice chairman of the powerful House Energy and Commerce committee.