On a sweeping patio overlooking the golf course at his private club in Bedminster, N.J., former President Donald J. Trump dined Sunday night with a close political ally, Marjorie Taylor Greene.

It was a chance for the former president to catch up with the hard-right Georgia congresswoman. But over halibut and Diet Cokes, Ms. Greene brought up an issue of considerable interest to Mr. Trump — the push by House Republicans to impeach his likely opponent in next year’s election.

“I did brief him on the strategy that I want to see laid out with impeachment,” Ms. Greene said in a brief phone interview.

Mr. Trump’s dinner with Ms. Greene came just two nights before Speaker Kevin McCarthy announced his decision on Tuesday to order the opening of an impeachment inquiry into President Biden, under intense pressure from his right flank.

Over the past several months, Mr. Trump has kept a close watch on House Republicans’ momentum toward impeaching Mr. Biden. Mr. Trump has talked regularly by phone with members of the ultraconservative House Freedom Caucus and other congressional Republicans who pushed for impeachment, according to a person close to Mr. Trump who was not authorized to publicly discuss the conversations. Mr. Trump has encouraged the effort both privately and publicly.

Ms. Greene, who has introduced articles of impeachment against Mr. Biden, said she told Mr. Trump that she wanted the impeachment inquiry to be “long and excruciatingly painful for Joe Biden.”

She would not say what Mr. Trump said in response, but she said her ultimate goal was to have a “long list of names” — people whom she claimed were co-conspirators involved in Biden family crimes. She said she was confident Mr. Trump would win back the White House in 2024 and that she wanted “to go after every single one of them and use the Department of Justice to prosecute them.”

While Mr. Biden’s son Hunter was charged in June with two misdemeanor tax offenses and a felony firearm offense, Republicans have not shown that Mr. Biden committed any crimes. House Republicans are proceeding with the impeachment inquiry without proof that Mr. Biden took official actions as vice president to benefit his son’s financial interests or that he directly profited from his son’s foreign deals.

Mr. Trump has also spoken weekly over the past month to Representative Elise Stefanik of New York, the third-ranking House Republican, according to a person familiar with the conversations who was not authorized to discuss them publicly. During those conversations, Ms. Stefanik also briefed Mr. Trump on the impeachment inquiry strategy, this person said.

The former president thanked Ms. Stefanik for publicly backing the impeachment inquiry in July, the person added. Ms. Stefanik, who talked to Mr. Trump again on Tuesday after Mr. McCarthy ordered the impeachment inquiry, had been the first member of House Republican leadership to publicly call for taking the first step in the process of impeaching Mr. Biden.

A person familiar with Mr. Trump’s thinking said that despite his eagerness to see an inquiry move forward, the former president has not been twisting Mr. McCarthy’s arm. Mr. Trump has been far more aggressive in pushing several members to wipe his own impeachment record clean, the person said, potentially by getting Congress to take the unprecedented step of expunging his two impeachments from the House record.

Mr. Trump has not been expressing concern about the possibility that the McCarthy impeachment effort might backfire and benefit Mr. Biden, according to two people with direct knowledge of his private statements over several months. Instead, he wondered to an ally why there had been no movement on impeaching Mr. Biden once he learned that the House was back in session.

A spokesman for Mr. McCarthy did not respond to a question about his interactions with the former president regarding impeachment.

When asked for comment, Mr. Trump’s communications director, Steven Cheung, pointed to Mr. Trump’s public statements about impeaching Mr. Biden.

The former president’s public commentary on the possibility of a Biden impeachment has escalated from wistful musings about the Justice Department’s supposed inaction to explicit demands.

“They persecuted us and yet Joe Biden is a stone-cold criminal, caught dead to right, and nothing happens to him. Forget the family. Nothing happens to him,” the former president said at a rally in March.

In a June town hall with the Fox News host Sean Hannity, Mr. Trump lamented what happened after authorities found boxes of classified documents in both Mar-a-Lago and the Bidens’ Delaware residence.

“It is a dual system of government,” Mr. Trump said. “You talk about law and order. You can’t have law and order in a country where you have such corruption.”

That same month, after Mr. Trump was arraigned on charges that he had improperly retained sensitive national security documents and obstructed investigators, he declared that if re-elected he would appoint a special prosecutor to “go after” Mr. Biden and his family.

By July, Mr. Trump had begun suggesting that Republicans should impeach the president, and as the summer wore on, he conveyed his desire with greater urgency.

“So, they impeach me over a ‘perfect’ phone call, and they don’t impeach Biden for being the most corrupt president in the history of the United States???” Mr. Trump wrote in all caps on his website, Truth Social.

In yet another nearly all-cap Truth Social post in late August, the former president wrote, referring to congressional Republicans: “Either impeach the bum, or fade into oblivion. They did it to us!”