WASHINGTON — President Trump continued his attacks on Attorney General Jeff Sessions on Tuesday, reviving his campaign call to investigate Hillary Clinton’s “crimes” as he criticized Mr. Sessions’s inaction.

In a news conference Tuesday, Mr. Trump also said he was “disappointed” in Mr. Sessions for recusing himself in the Justice Department’s investigation into the Trump campaign’s ties with Russia.

“He should not have recused himself almost immediately after he took office, and if he was going to recuse himself, he should have told me prior to taking office, and I would have, quite simply, picked somebody else,” Mr. Trump said. “So I think that’s a bad thing, not for the president but for the presidency.”

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Pressed to say whether he intends to fire Mr. Sessions, the president said he wants the attorney general to be “much tougher” on leaks from intelligence agencies. But he declined to say whether he wanted Mr. Sessions to resign.

“I told you before, I’m very disappointed in the attorney general,” he said. “We will see what happens. Time will tell. Time will tell.”

Two Twitter posts Tuesday morning were the latest in a string of attacks on Mr. Sessions that began when he told The New York Times in an interview that he never would have appointed Mr. Sessions if he had known he would recuse himself in the Russia inquiry. The recusal was the first in a series of steps that led to the appointment of a special counsel to oversee that investigation.

One of Mr. Trump’s top advisers acknowledged that the president most likely wanted Mr. Sessions out of the attorney general’s office.

The adviser, Anthony Scaramucci, hired last week as White House communications director, said he did not want to speak for Mr. Trump, but given the level of public tension between the president and his attorney general, it’s “probably right” that Mr. Trump wants him out of that job.

“He’s obviously frustrated,” Mr. Scaramucci said in an interview with the conservative radio host Hugh Hewitt.

On Monday, Mr. Trump called Mr. Sessions “beleaguered” in a tweet, raising questions about whether Mr. Sessions would resign. Mr. Sessions, one of Mr. Trump’s earliest campaign supporters, has previously shown no indication that he was considering resigning.

Mr. Trump was critical on Tuesday about what he called Mr. Sessions’s “VERY weak position” on an investigation into Mrs. Clinton’s use of a private computer server. The F.B.I. investigated and closed the inquiry without charges in 2016.

Mr. Scaramucci told CNN on Monday that Mr. Sessions and the president needed to “sit down, face to face, and have a reconciliation and a discussion of the future.”

If Mr. Sessions were to resign or be fired, Mr. Trump could appoint a successor during the congressional recess who would not face Senate inquiries into his or her position on recusal and could take over, at least temporarily, without a confirmation vote. That could allow the president to assert greater control over the special counsel investigation into his campaign’s contacts with Russia.

In another tweet on Tuesday, Mr. Trump attacked the acting F.B.I. director, Andrew McCabe, for what Mr. Trump described as his role in the investigation into Mrs. Clinton’s computer use.

In criticizing Mr. McCabe, Mr. Trump revived reports that Mr. McCabe’s wife accepted contributions from a longtime Clinton supporter in her bid for a Virginia state Senate seat.

Mr. Trump also tweeted that Ukraine tried to “sabotage” his presidential campaign and help Mrs. Clinton.

Mr. Trump appeared to be referencing the Fox News host Sean Hannity’s discussion of a Politico article in January about Mrs. Clinton’s allies coordinating with Ukrainian officials to research politically harmful information about Mr. Trump and his advisers. Mr. Trump’s supporters have used this anecdote to justify the actions of Mr. Trump’s eldest son, Donald J. Trump Jr. In June 2016, the younger Mr. Trump had a meeting with a Russian lawyer in which he was promised potentially damaging information on Mrs. Clinton.

United States intelligence agencies have concluded that Russia hacked into Democratic servers and stole emails in an effort to help Trump win the election.