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WASHINGTON— Rep. Matt Gaetz (R., Fla.) on Friday said he wouldn’t resign in the face of rising pressure over reports that the FBI has been investigating him since last summer, in part over whether he broke federal sex-trafficking laws.

"Of course not," Mr. Gaetz said in a text message Friday, addressing whether he would leave office. He declined to say more, citing Good Friday. His lawyer also said Mr. Gaetz wouldn’t step down.

With the House in recess, members have been largely silent about Mr. Gaetz since Minority Leader Kevin McCarthy (R., Calif.) earlier in the week referred to "serious implications" in connection with the Federal Bureau of Investigation inquiry. But in the state capital of Tallahassee, Republicans were already discussing who would replace Mr. Gaetz, people familiar with the conversations said.

"His political brethren are already jockeying to take his seat," said former Florida Rep. David Jolly. "They’re not insulating him—they’re already assuming that Matt’s political career is over."

"These scandals hit a certain point where there’s no escape," Mr. Jolly said. "We’ve clearly hit that point for Matt in Florida politics."

"These scandals hit a certain point where there’s no escape. We’ve clearly hit that point for Matt in Florida politics."

— David Jolly, former Florida congressman
David Jolly speaks Oct. 20, 2018, in Los Angeles. (Getty Images)

David Jolly speaks Oct. 20, 2018, in Los Angeles. (Getty Images)

The Justice Department’s investigation is examining whether Mr. Gaetz and another man, former Seminole County tax collector Joel Greenberg, paid women in cash or gifts in exchange for sex, people familiar with the investigation said.

Investigators believe Mr. Greenberg, who was indicted last year on a federal sex-trafficking charge and other crimes, met the women online through so-called sugar-daddy websites that connect people for dates in exchange for expensive gifts or travel. Mr. Greenberg has pleaded not guilty to charges including illegally obtaining personal information from a state computer database, wire fraud and money laundering. He is expected to face trial in June.

The investigation is looking at whether Mr. Greenberg introduced some of the women to Mr. Gaetz, who also allegedly had sex with them, the people said. People familiar with the probe have described Mr. Greenberg as a friend and political ally of the congressman. The connection was earlier reported by the New York Times.

The Justice Department is also investigating whether Mr. Gaetz had sex with a girl who was 17 at the time of the encounters about two years ago, The Wall Street Journal has reported. The probe is looking at whether she received anything of value in exchange, including travel, in violation of federal sex-trafficking laws.

Investigators in recent months have interviewed multiple women as part of the case, the people said.

Mr. Gaetz has denied any wrongdoing and portrayed the accusations against him as part of a scheme to get his family to pay $25 million to put an end to the investigation. "What is happening is an extortion of me and my family involving a former Department of Justice official," he said earlier this week.

Many House Republicans seek an early end to the attention to Mr. Gaetz because they don’t want it to damage what they believe is a chance to take back control of the House in 2022 given the Democrats’ narrow 219-211 majority. Privately, many Republicans in Congress are ready for Mr. Gaetz to leave, according to GOP figures who have spoken to lawmakers.

Mr. Gaetz’s behavior in Congress has raised eyebrows among some lawmakers and aides on the Hill. One former lawmaker described how Mr. Gaetz had shown him videos of his sexual partners in the House Republican cloakroom. A current lawmaker described Mr. Gaetz one day bragging to him on a walk from the Capitol about going to bars and being around "fine women." A congressional aide said that after Mr. Gaetz got engaged last year, he said, "I’m really loving this monogamy thing," adding that Mr. Gaetz was drawing a contrast to his previous lifestyle.

Mr. Gaetz didn’t respond to a further request for comment Friday evening.

Mr. Gaetz’s behavior also came into question during his service in Florida’s state legislature from 2010 to 2016, according to a person familiar with the matter. While in Tallahassee, he and others competed against each other in a contest over having sexual relationships with women, operating under a point system in which participants were awarded one point for sleeping with a lobbyist and two points if the lobbyist was married, this person said.

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"You created a game where members of the FL House got "points" for sleeping with aides, interns, lobbyists, and married legislators," said Florida state Rep. Chris Latvala and fellow Republican on Twitter last year. "Hope DC is treating you well."

At the time Mr. Gaetz replied on Twitter that "just because I own you on twitter, don’t confuse me for your daddy when it comes to abusing power for sex," in a reference to Mr. Latvala’s father, a powerful state lawmaker who resigned in 2018 after facing accusations of sexual harassment.