Trump, and other Republican ’24 contenders, push claims of injustice.

The Justice Department would charge Mr. Biden but agree not to prosecute him in connection with his purchase of a handgun in 2018 during a period when he was using drugs. The deal would be contingent on Mr. Biden remaining drug free for 24 months and agreeing never to own a firearm again.

The agreement must still be approved by a federal judge. Mr. Biden is expected to appear in federal court in Delaware in the coming days to be arraigned on the misdemeanor tax charges and plead guilty.

“With the announcement of two agreements between my client, Hunter Biden, and the United States Attorney’s Office for the District of Delaware, it is my understanding that the five-year investigation into Hunter is resolved,” Mr. Biden’s lawyer, Christopher Clark, said in a statement.

A White House spokesman, Ian Sams, said in a statement, “The president and first lady love their son and support him as he continues to rebuild his life. We will have no further comment.”

Assuming there are no last-minute changes or complications, the deal would most likely resolve the investigation without Mr. Biden facing a federal prison sentence.

But it would by no means end the superheated politics of the case. Republicans have sought for years to make the case that Hunter Biden committed an array of crimes that should put him behind bars and call into question the honesty of his father.

Coming less than two weeks after the Justice Department indicted former President Donald J. Trump on charges that he risked exposing national security secrets and obstructed efforts by the government to reclaim classified documents from him, an agreement that allows Hunter Biden to walk free is also sure to bring a torrent of criticism from the right and intensified efforts by House Republicans to portray the Justice Department and the F.B.I. as biased.

As president, Mr. Trump had long sought to tie Hunter Biden’s business deals and personal troubles to his father. Mr. Trump’s first impeachment had its roots in his efforts to persuade the Ukrainian government to help him show wrongdoing in Hunter Biden’s work for Burisma, a Ukrainian energy company, and while in the White House he pressured the Justice Department to investigate.

The Justice Department investigation continued after President Biden took office, under the oversight of the U.S. attorney for Delaware, David C. Weiss, a Trump appointee who was kept on to allow him to finish the inquiry. Attorney General Merrick B. Garland has testified to Congress that Mr. Weiss had full authority and independence to decide whether to bring a case against Mr. Biden.

Along with resolving the main legal issues confronting Hunter Biden, the agreement is a victory for his lead lawyer, Mr. Clark, a hard-charging former federal prosecutor. In meetings with Justice Department officials over the past year and a half, he has presented them with evidence intended to convince them that any prosecution of his client would be weak.

“Hunter will take responsibility for two instances of misdemeanor failure to file tax payments when due pursuant to a plea agreement,” Mr. Clark said in his statement. “A firearm charge, which will be subject to a pretrial diversion agreement and will not be the subject of the plea agreement, will also be filed by the government. I know Hunter believes it is important to take responsibility for these mistakes he made during a period of turmoil and addiction in his life. He looks forward to continuing his recovery and moving forward.”

The investigation focused on a particularly chaotic and unseemly period in Hunter Biden’s life when he was addicted to crack cocaine. But the Justice Department went through nearly every major aspect of his life over the past 15 years — a period in which he also struggled to control his alcoholism and engaged in international business deals, which he got into at least in part because of his father’s prominence in politics.

And despite federal investigators casting their nets wide — including examining Mr. Biden’s work for Burisma and his business dealings with an energy tycoon in China — the investigation ultimately narrowed to two separate issues.

One was his taxes. Prosecutors had been weighing whether to indict him in connection with his failure to meet filing deadlines for his 2017 and 2018 taxes, and whether he had improperly claimed $30,000 in deductions for business expenses.

The second was whether he lied on a United States government form that he filled out when he purchased the handgun in 2018. In response to a question on the form about whether he was using drugs, Mr. Biden had said he was not — an assertion that prosecutors suspected might be false based on his erratic behavior at the time and accounts from people who interacted with him.

Those charges were far less explosive than ones pushed by Mr. Trump and congressional Republicans, who have been angry with the Justice Department for failing to find wider criminal wrongdoing by the president’s son and family.

Since taking control of the House in January, top Republicans have used their new investigative power to push the narrative that the president has been complicit in an effort engineered by his son to enrich his family by profiting from their positions of power.

No one questions that Mr. Biden, a 53-year-old Yale-educated lawyer, has had significant personal troubles and pursued a professional path that has intersected with his father’s in ways that raised ethical issues.

After his father became vice president, he built relationships with wealthy foreigners that brought in millions of dollars, surfacing concerns inside the Obama administration and among government watchdog groups that he was cashing in on his family name.

He went into a downward spiral after his brother, Beau, died in 2015, becoming addicted to crack and engaging in tawdry, self-destructive behavior.

But the questions about what occurred during that period never led to conduct that prosecutors believed could win them a conviction in court.

In Ukraine, Hunter Biden sat on the board of Burisma, which was led by an oligarch who at the time was under investigation for corruption. He was paid tens of thousands of dollars a months for the position, which he held while his father was vice president and overseeing the Obama administration’s Ukraine policy.

Republicans have also pointed to an equity stake that Hunter Biden took in a Chinese business venture, and to his failed joint venture with a Chinese tycoon who had courted well-connected Americans in both major parties — at one point he gave the younger Mr. Biden a large diamond as a gift — but who was later detained by Chinese authorities.

Allegations promoted by Republicans that the elder Mr. Biden’s Justice Department went easy on his son are unlikely to fade away.

In April, an I.R.S. supervisor who had been overseeing the investigation into Hunter Biden hired a lawyer and went to Congress, alleging political favoritism in how the investigation had been handled. Congressional Republicans have pledged to investigate the claims, which have also been referred to inspectors general at the Justice Department and I.R.S.

Seamus Hughes contributed reporting.