The White House insisted on Friday that House Republicans end their effort to impeach President Biden, declaring that “enough is enough” after their monthslong inquiry failed to turn up promised evidence of high crimes and misdemeanors.

“It is obviously time to move on, Mr. Speaker,” Edward N. Siskel, the White House counsel, wrote in a four-page letter to Speaker Mike Johnson. “This impeachment is over. There is too much important work to be done for the American people to continue wasting time on this charade.”

The letter comes as the Republican impeachment drive has all but collapsed after the indictment of a key witness on charges of making up allegations against Hunter Biden, the president’s son. A number of Republicans have cast doubt on the venture, and even some champions of impeachment have now concluded that they could not muster a majority if they sent articles to the floor charging Mr. Biden.

The White House hopes to capitalize on Republicans’ disarray, in effect calling their bluff and daring them to put up or shut up, although the hard-liners in the G.O.P. conference are unlikely to choose either option. Mr. Biden’s team harbors little hope that Republicans will formally call off the inquiry, much less acknowledge that they have nothing much to show for it, but the president’s advisers want to put a punctuation mark on the Republican setbacks and make clear to the public that impeachment is effectively dead.

It is part of a newly aggressive strategy by the president as he embarks on his re-election campaign in earnest, starting with his confrontational State of the Union address last week and his active schedule of travel in battleground states since. After a period in which allies feared Mr. Biden was being too passive, he hopes to get back on offense as he engages in a rematch with former President Donald J. Trump, whom he defeated in 2020.

House Republicans are not quite ready to give up. They argue that they are still investigating and have scheduled a hearing with Hunter Biden’s former business associates next week. They are also demanding recordings from the investigation of the special counsel Robert K. Hur, who examined Mr. Biden’s handling of classified documents, even though that was not among the topics of the impeachment inquiry and Mr. Hur decided no criminal charges were warranted.

But in a recognition that an impeachment vote is unlikely at this point, Republicans have been exploring an alternative strategy of issuing criminal referrals urging the Justice Department to investigate Mr. Biden or people around him. Such a move would carry no legal weight and would essentially be little more than a symbolic statement, unless Mr. Trump wins and uses the referrals to justify a prosecution of Mr. Biden after he leaves office.

Unsurprisingly, Mr. Johnson did not take the advice from the White House. “It is not surprising that the White House would prefer to close the ongoing House inquiry which has uncovered that the Biden family and their associates received over $20 million from foreign sources, and that President Biden has lied repeatedly,” said Raj Shah, a spokesman for the speaker. “The White House does not get to decide how impeachment gets resolved, that is for Congress to decide.”

While Republicans say that the Bidens and their circle made more than $20 million from foreign sources in China, Ukraine and elsewhere, fact checkers have concluded that much of the money went to their business associates, not the family itself, and what did go to the Bidens was payment to Hunter Biden and the president’s brother James Biden, who had engaged in business deals.

Republicans have failed to show that Mr. Biden received any profit, and so far have only pointed to his son and brother paying him back for a few no-interest loans. Republicans also have produced no evidence that the business relationships were illegal or resulted in a quid pro quo by the president.

In his letter on Friday, Mr. Siskel needled the House G.O.P. majority over its problems with impeachment. He quoted Republicans themselves as saying that they “can’t identify a particular crime” supposedly committed by the president and lamenting that they had made impeachment “a social media issue as opposed to a constitutional concept.”

“The House majority ought to work with the president on our economy, national security and other important priorities on behalf of the American people, not continue to waste time on political stunts like this,” Mr. Siskel wrote.

Rather than finding proof that the president committed impeachable offenses, he added, “the investigation has continually turned up evidence that, in fact, the president did nothing wrong.” He listed 20 witnesses whose testimony, in his view, undercut the Republican theory that money paid to Hunter Biden by foreign firms amounted to bribery and noted that “the majority cannot identify any policy or governing decisions that were supposedly improperly influenced.”

Mr. Siskel criticized Republicans for seeking to interview again witnesses who had already testified, “perhaps hoping the facts will be different the second time around,” which he called “just the latest abusive tactic in this investigation.” Republican efforts to seize on Mr. Hur’s investigation, he added, amounted to searching for “a flotation device for the sinking impeachment effort.”

The Republican investigation began shortly after the party took control of the House early last year and was authorized as an official impeachment inquiry in September by Kevin McCarthy, the House speaker at the time. The full House voted in December to formalize the inquiry on a strictly party-line vote.

The impeachment drive, however, took a major blow last month when Alexander Smirnov, a witness relied on by the Republicans, was charged with fabricating claims that Mr. Biden and his son had each sought $5 million bribes from a Ukrainian company. Mr. Smirnov told investigators that “officials associated with Russian intelligence were involved in passing a story” about Hunter Biden.

“None of the evidence has demonstrated that the president did anything wrong,” Mr. Siskel wrote. “In fact, it has shown the opposite of what House Republicans have claimed.”