But the president’s options are constrained, advisers said, because he recognizes that he would have a difficult time winning Senate confirmation for a replacement. Mr. Sessions served there for 20 years, and his former colleagues have bristled at Mr. Trump’s attacks. Any dismissal of Mr. Sessions would be taken by Democrats and even some Republicans as an effort to seize control of the Russia investigation and could trigger a bipartisan backlash.
The exchange on Wednesday began when the president lashed out at Mr. Sessions for seeming to suggest that the Justice Department’s inspector general would look into Republican charges of misconduct in the opening stage of the Russia investigation rather than opening his own examination.
“Why is A.G. Jeff Sessions asking the Inspector General to investigate potentially massive FISA abuse,” Mr. Trump wrote on Twitter. “Will take forever, has no prosecutorial power and already late with reports on Comey etc. Isn’t the I.G. an Obama guy? Why not use Justice Department lawyers? DISGRACEFUL!”
Republicans have accused Justice Department and F.B.I. officials of abusing their powers while President Barack Obama was still in office by using information from a dossier prepared by a former British spy paid by Hillary Clinton’s 2016 presidential campaign to justify surveillance of a former Trump campaign adviser, Carter Page. Officials did not fully inform the court that issues warrants under the Foreign Intelligence Surveillance Act, or FISA, of the origin of the information, Republicans complained. Democrats have called that a distortion and distraction.
The inspector general, Michael E. Horowitz, was appointed by Mr. Obama in 2012, but previously worked for the Justice Department under Republican and Democratic presidents. He has already been investigating how James B. Comey, the F.B.I. director until Mr. Trump fired him last spring, handled the inquiry into Mrs. Clinton’s use of a private email server while she was secretary of state.
Mr. Sessions seemed to take umbrage at the president’s latest message. “We have initiated the appropriate process that will ensure complaints against this department will be fully and fairly acted upon if necessary,” he said in his statement.
“As long as I am the attorney general,” he added, “I will continue to discharge my duties with integrity and honor, and this department will continue to do its work in a fair and impartial manner according to the law and Constitution.”
Mr. Sessions’s response, polite but pointed, was all the more striking because he had largely kept quiet after previous attacks by the president. Mr. Trump has never forgiven Mr. Sessions for recusing himself from the Russia investigation, a decision that helped lead to the appointment of Mr. Mueller. Mr. Trump has publicly called Mr. Sessions “weak” and said he would not have appointed him had he known Mr. Sessions would step aside.
His latest eruption was prompted by Mr. Sessions’s comment on Tuesday that if a FISA surveillance warrant was wrongfully obtained, the matter would be “investigated” by the department’s inspector general. His comment was interpreted as confirmation that the inspector general had opened a second official inquiry on top of the Comey review.
But Mr. Sessions only meant to reiterate what he said after a memo drafted by House Republicans was released alleging abuse of the FISA process. At the time, Mr. Sessions said he would “forward to appropriate D.O.J. components all information I receive from Congress regarding this.”
The nuance was lost on Mr. Trump, who among other things did not seem to understand that an attorney general cannot order an inspector general to investigate anything, only refer information.
“The president’s tweet reveals that he really doesn’t understand how the government works and how the Justice Department works,” said Michael Bromwich, a former department inspector general.
He added that the inspector general’s office has a reputation for professionalism. “It’s incredibly demoralizing to have the chief executive of the government not only not understand and appreciate what you do, but attack what you do on a constant basis,” Mr. Bromwich said.
Inspectors general at cabinet agencies are kept separate to preserve their independence. Paul Light, a New York University professor and specialist on the offices, recalled that President Ronald Reagan fired all of the inspectors general but was forced by Congress to rehire some of them. “They have protections in statutes against arbitrary dismissal,” he said.
After Mr. Trump’s tweet, Mr. Horowitz, the inspector general, received support from Republicans, including Senator Charles E. Grassley of Iowa, the chairman of the Senate Judiciary Committee that confirmed him.
“I have complete confidence in him and hope he is given the time, the resources and the independence to complete his work,” said Representative Trey Gowdy, Republican of South Carolina and the chairman of the House Oversight Committee.
Several Republicans expressed dismay at the president’s continued campaign against Mr. Sessions. “It’s kind of mind-boggling that he would call out his own attorney general,” former Representative Jason Chaffetz of Utah said on Fox News.
Representative Peter T. King of New York, also on Fox, expressed sympathy with Mr. Trump’s desire for a second investigation run by Mr. Sessions, but added that the president should not berate him. “Jeff Sessions is loyal to the president and he’s one of the first to support him, and he’s often in very difficult positions and I think he’s trying to reconcile as best as he can,” he said.
Michael W. McConnell, a former appellate judge now at Stanford Law School, said a president has every right to direct his attorney general.
“What raises eyebrows is the form and tone of the tweet, which appears to be a commentary on the attorney general’s decisions rather than an exercise of presidential supervisory authority,” he said. “Mr. Trump is the president. If he wants something done differently, he should order that it be done differently, with serious reflection, through proper channels and in the proper form.”
Jamil Jaffer, a law professor at George Mason University and former associate White House counsel under President George W. Bush, said social media was not the best way to direct action by an attorney general. “The president has a lot of tools that are a lot more effective than putting the A.G. on blast on Twitter,” he said.
But Mr. Trump got support from other quarters. Representative Lee Zeldin of New York and a dozen other Republicans sent a letter to Mr. Sessions on Wednesday urging him to appoint a special counsel to investigate the handling of Mrs. Clinton’s case and the FISA warrant targeting Mr. Page.
The Rev. Jerry Falwell Jr., the president of Liberty University, suggested that Mr. Sessions had never really supported Mr. Trump in the first place.
“@USAGSessions must be part of the Bush/Romney/McCain Republican Establishment,” he wrote on Twitter. “He probably supported @realDonaldTrump early in campaign to hide who he really is. Or he could just be a coward.”