President Trump on Saturday excoriated Attorney General William P. Barr, castigating him on Twitter for not violating Justice Department policy to publicly reveal an investigation into President-Elect Joseph R. Biden Jr.’s son.

The critical tweets about Mr. Barr, who has largely been a close confidant to the president since he was appointed two years ago, came a day after the Supreme Court rejected a lawsuit seeking to subvert the results of the election. With the Electoral College set to meet on Monday and Congress to formally tally the results in January, the prospects for Mr. Trump to change the outcome are all but gone.

The president’s statements undermining faith in the electoral process — and his assaults on institutions — have escalated since the election on Nov. 3, as he enters the final weeks of his time in office. Privately, he has railed against Mr. Barr for not bolstering his false claims of widespread fraud in the election and instead affirming Mr. Biden’s victory.

His messages on Saturday echoed his attacks on his first attorney general, Jeff Sessions, whom he blamed for recusing himself from overseeing the investigation into whether the Trump campaign had colluded with Russian officials in the 2016 election. For months, Mr. Trump publicly berated Mr. Sessions before firing him in November 2018, a day after the midterm races.

In three tweets, Mr. Trump called the attorney general a “big disappointment” and denounced him for not disclosing the existence of an investigation into Hunter Biden for possible tax evasion, which he said would have given Republicans an edge in the election. Doing so would have violated department guidelines about publicly discussing ongoing cases. Mr. Trump benefited from that policy himself in 2016, when officials kept quiet the inquiry into possible conspiracy between his campaign and Russian officials.

“Why didn’t Bill Barr reveal the truth to the public, before the Election, about Hunter Biden,” Mr. Trump wrote. “Joe was lying on the debate stage that nothing was wrong, or going on – Press confirmed. Big disadvantage for Republicans at the polls!”

The president has told aides he would like to see Mr. Barr appoint a special counsel to investigate the younger Mr. Biden, according to people briefed on the discussions. He has not expressed that desire directly to Mr. Barr, according to a person familiar with the conversations, but has instead let the issue become public in the hope of creating a pressure campaign.

Mr. Barr is so far unlikely to appoint such a special prosecutor, according to people familiar with the thinking. The question remains whether Mr. Trump will succeed in forcing him to resign or will fire him so he can appoint someone willing to do the president’s personal bidding.

The president’s interest in appointing a special counsel was earlier reported by The Wall Street Journal.

A spokeswoman for the Justice Department declined to comment on the president’s tweets.

Mr. Trump did not answer questions from reporters as he left the White House around noon to travel to West Point for the annual Army-Navy football game on Saturday. He has mostly stayed out of sight since Election Day, taking few questions from journalists and attending only a handful of public events.

Last Sunday, The New York Times reported that Mr. Barr was considering resigning before the end of the term, a decision that he had been weighing for weeks. The attorney general was convinced Mr. Trump had lost the election, believed his work at the Justice Department was completed and wanted to avoid the controversy that often comes at the end of an administration.

In response to the reporting, some Republicans lobbied Mr. Barr to reconsider his plans, and the attorney general let the White House know that he intended to stay through the end of the term.

After Hunter Biden disclosed on Wednesday that the Justice Department was investigating his taxes, the president’s anger toward his attorney general grew.

Mr. Barr has long been considered a close ally of the president. His public summary of the lengthy report by Robert S. Mueller III, the special counsel who was appointed to investigate Russian interference, cast the contents in a favorable light for Mr. Trump, drawing protests from Mr. Mueller himself.

Mr. Barr also worked with the White House counsel, Pat A. Cipollone, to publicly release the transcript of the call that Mr. Trump had with the president of Ukraine in July 2019. In that call, the president — who was withholding congressionally approved military aid to Ukraine — pushed for investigations into the Bidens.

But their relationship has come under strain this year, with the president and Mr. Barr speaking infrequently. In February, as Mr. Trump widened his attacks on law enforcement, Mr. Barr publicly rebuked the president, saying that Mr. Trump’s tweets had made it “impossible” for him to do his job.

In the weeks after the election, Mr. Barr refused to refute Mr. Trump’s specious claims of widespread voter fraud. But this month, after Mr. Trump raised the prospect that the Justice Department and F.B.I. may have been involved in tipping the election to Mr. Biden, Mr. Barr broke his silence. In an interview with The Associated Press, Mr. Barr said that he saw no examples of widespread voter fraud that could have meaningfully affected the election.

Those comments angered Mr. Trump, who has been searching for anyone to help push the notion that the election was stolen from him.

Days before the election, the Justice Department announced that Mr. Barr had appointed a top federal prosecutor as a special counsel to examine how the F.B.I. and intelligence agencies investigated the ties between Mr. Trump’s campaign and Russia.

The announcement miffed Mr. Trump, who had wanted Mr. Barr to make public such a disclosure before the election, when Mr. Trump could have weaponized it on the campaign trail, as he did with the federal investigation into Hillary Clinton’s email server and the release of stolen emails from her campaign chairman aired publicly through WikiLeaks.

Also in Washington on Saturday, a small group of the president’s supporters joined a “Stop the Steal” march. Among those present was Enrique Tarrio, a leader of the far-right group the Proud Boys who also led a “Latinos for Trump” effort during the presidential campaign.

On the social media site Parler, Mr. Tarrio posted pictures of himself at the White House and said he had a “last-minute” invitation. Judd Deere, a spokesman for the White House, said Mr. Tarrio “was on a public White House Christmas tour” but did not meet with the president nor had the White House invited him.