Trump nemesis Michael Avenatti announced Tuesday that he will not seek Democrats’ nomination to challenge President Trump in 2020, saying he feared the damage to his family.

“But for their concerns, I would run,” said the California attorney whose legal crusades against the president and his agenda have foundered in recent months.

He did not say what his family’s concerns were but said they were the ones to “request” he forgo a run.

He signaled, though, that he is not about to retire from being a thorn in Mr. Trump’s side, with clients who want to take on the president and hinder his agenda.

“I will continue to represent Stormy Daniels and others against Donald Trump and his cronies and will not rest until Trump is removed from office, and our republic and its values are restored,” he said in a statement.

And he said he was “concerned” that Democrats aren’t prepared to be nasty enough to fight back against Mr. Trump in 2020, saying he fears the party will nominate someone capable of being a good president, “but has no chance of actually beating Donald Trump.”

“We will not prevail in 2020 without a fighter. I remain hopeful the party finds one,” he said.

Mr. Avenatti’s bid had not gained much traction, according to polls that showed him in the low single digits among potential Democratic primary voters.

But he had taken his own bid seriously, making the rounds of early primary states and scoring invites to party events.

Some of those began to dry up in recent weeks, however, after he was arrested in Los Angeles on domestic violence allegations and suffered other setbacks.

Ms. Daniels, an adult film actress, had her defamation lawsuit against Mr. Trump tossed by a court, with the judge also ordering her to pay Mr. Trump’s lawyers’ fees. Those lawyers submitted a bill for $389,503.11 to the court this week.

The Senate Judiciary Committee chairman has also asked the federal Justice Department to investigate Mr. Avenatti and another client for obstructing and lying to Congress after the client made unsubstantiated and conflicting claims of rape to influence the confirmation of Supreme Court Justice Brett M. Kavanaugh.

Mr. Avenatti in his statement announcing his decision did not specify who in his family had told him not to run. He is twice-divorced and has children from both marriages.

He had been polling at just a percentage point or two among potential Democratic primary voters, or well behind the front-runners such as former Vice President Joseph R. Biden.

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