On the big screen, it comes off as a scene of high drama: an icon of the civil rights movement upbraiding a hesitant president in the Oval Office, as a portrait of George Washington bears mute witness.

“Mr. President, in the South, there have been thousands of racially motivated murders,” Martin Luther King Jr. says, imploring Lyndon B. Johnson to put his weight behind ensuring voting rights for black Americans. “We need your help!”

To which he gets a pat on the shoulder. “Dr. King, this thing’s just going to have to wait,” Johnson says.

In real life, that December 1964 meeting happened — but not that way, according to one who was there.

“It was not very tense at all. We were very much welcomed by President Johnson,” recalled former Atlanta mayor and U.N. ambassador Andrew Young, who attended the session as a young lieutenant to King. “He and Martin never had that kind of confrontation.”

As Oscar buzz builds for “Selma,” the new movie about a pivotal episode in the civil rights movement is attracting controversy over its portrayal of King and Johnson’s complicated relationship.

Former Johnson aides have engaged in a war of words with director Ava DuVernay over the film’s depiction of the president as an early adversary of King’s push for legislation that would guarantee that blacks be allowed to exercise their constitutional right to vote — something that officials in many Southern counties prevented in practice by imposing discriminatory literacy tests, poll taxes and other obstacles.

The argument over “Selma’s” accuracy is far from the first time that a historically themed movie has been accused of blurring the line between art and fact. But this dispute comes as some of the themes of “Selma” are being replayed in today’s headlines and as protests have erupted in the wake of incidents in which African Americans have died at the hands of law enforcement.

A mountain of available records suggests that there are ­places where the director and her critics are each right — and ones where each is wrong.

That is because Johnson and King’s relationship was a partnership of two shrewd Southerners who had common goals but sometimes conflicting priorities. And the unspooling of events a half-century later can look very different, depending on whether it is viewed from the perspective of Washington deal-making or from the dangerous front lines.

As the movie depicts, the long-simmering issue of voting rights came to a head with three 1965 marches in Selma, Ala. — the first of which, on March 7, turned violent when local police beat and tear-gassed peaceful civil rights demonstrators in front of network television cameras. That shocking sight galvanized public opinion and provided Johnson an opportunity to put forward the Voting Rights Act of 1965.

With its focus on King and the foot soldiers of the civil rights movement, “Selma” is a departure from other big-budget movie treatments of that chaotic period in U.S. history. Earlier films — including “To Kill a Mockingbird,” “Mississippi Burning” and “The Help” — have often portrayed black characters as helpless until a white champion emerges to defend and mobilize them.

But those who are close to Johnson and his legacy argue the film goes too far in the other direction, sidelining the late president and painting him in an unfairly harsh light.

“ ‘Selma’s’ obstructionist LBJ is devoid of any palpable conviction on voting rights. Vainglorious and power hungry, he unleashes his zealous pit bull, FBI chief J. Edgar Hoover, on King,” Mark Updegrove, the director of Johnson’s presidential library, wrote in a Dec. 22 essay for Politico Magazine.

“At a time when racial tension is once again high, from Ferguson to Brooklyn, it does no good to bastardize one of the most hallowed chapters in the Civil Rights Movement by suggesting that the President himself stood in the way of progress,” Updegrove added.

Joseph A. Califano Jr., who was Johnson’s domestic policy chief, echoed that criticism in a Dec. 26 column in The Washington Post and argued that the movie’s depiction of the 36th president is so flawed that “Selma” should be “ruled out this Christmas and during the ensuing awards season.”

Both Updegrove and Califano cited a taped telephone conversation between King and Johnson on Jan. 15, 1965, in which the president suggests a strategy similar to the one King ultimately pursued in Selma. Califano went so far as to declare that “Selma was LBJ’s idea.”

Johnson advised King: “Find the worst condition that you run into in Alabama, Mississippi, or Louisiana, or South Carolina. . . . Take that one illustration and get it on radio and get it on television.”

DuVernay, who has received glowing critical reviews for her third feature film directorial effort and has become the first African American woman nominated for a Golden Globe for best director of a motion picture, shot back on Twitter on Sunday. She accused her critics of being the ones trying to distort history

DuVernay tweeted: “Notion that Selma was LBJ’s idea is jaw dropping and offensive to [civil rights groups] and black citizens who made it so.”

Some of the confusion may lie in King’s decision to play his cards close in his dealings with Johnson.

Weeks before his conversation with Johnson, King and his allies had zeroed in on Selma as a spot to make their stand — a plan that the civil rights leader did not share with the president when LBJ suggested a similar approach.

“King, on his heels, had mumbled approval. He did not mention that he was headed to Selma for that very purpose — knowing that Johnson would not welcome his tactics of street protest,” Pulitzer Prize-winning historian Taylor Branch wrote in “At Canaan’s Edge,” the third book in his King-era trilogy. (Branch, who has his own film project in the works, declined to comment on the “Selma” dispute.)

Johnson’s daughters, Lynda Bird Johnson Robb and Luci Baines Johnson, are furious about how their father is portrayed in the film, according to several sources.

For many years, Johnson’s legacy had been defined by the shadow of the Vietnam War, but more recently, his presidency’s accomplishments have gained new attention. The movie was released eight months after a celebration of the 50th anniversary of the Civil Rights Act of 1964 at the Johnson presidential library. President Obama and every living ex-president with the exception of the ailing George H.W. Bush showed up to pay tribute to Johnson.

So why does it matter whether a movie gets some things wrong about LBJ?

“Let me tell you how important it is,” Califano said in an interview. “Many, many of our young people get their view of history from films and television. It’s important for people who make movies that claim to be historically accurate to be accurate.”

The moment in the film that sets the tension between Johnson and King centers on the December 1964 meeting in the Oval Office. Johnson counters that he cannot push for a voting rights law so soon after the passage of the Civil Rights Act of 1964 — which required summoning all his legendary political skills to help break a 60-day filibuster in the Senate — and as he is trying to get his other Great Society programs off the ground.

Young recalled the real-life meeting in an interview with The Post in a three-way phone conversation in which DuVernay was also on the line. She declined to be interviewed on the record for this article.

Young lavished praise on her film, in which he shows up frequently as a character. Its depiction of the interaction between King and Johnson “was the only thing I would question in the movie. Everything else, they got 100 percent right,” Young said.

Johnson and King “were always mutually respectful. Martin respected Lyndon Johnson’s political problems,” Young said.

Historians, however, say there were frequent tensions. Branch, for instance, wrote that “Johnson in the White House was intensely personal but unpredictable — treating King variously to a Texas bear hug of shared dreams or a towering, wounded snit.”

On the other hand, accounts at the time also suggest that Johnson was willing to put his political capital on the line for voting rights, even before the Selma marches.

A front-page story in The Post on Jan. 6, 1965 — more than two months before the “Bloody Sunday” confrontation on the Edmund Pettus Bridge in Selma — reported that Johnson planned to submit a constitutional amendment that would ban literacy tests. He also was considering stopgap legislation that would provide federal voter registrars in place of the local officials who were preventing blacks from registering in the Deep South.

Johnson aides also dispute the film’s implication that the president was complicit in a scheme by Hoover to disrupt King’s home life by sending his wife, Coretta Scott King, an audiotape of her husband having sex with another woman.

“I’m 100 percent sure that was not true,” said Larry Temple, who was special counsel to Johnson in 1967 and 1968 and is now chairman of the foundation that supports his presidential library.

However they got there, Johnson and King were in complete accord on voting rights after Selma. In what is now regarded as one of the most powerful presidential speeches in history, Johnson used a rallying cry from an anthem of black protest — “We shall overcome” — to make the case in a nationally televised address for what would become the Voting Rights Act of 1965.

In his memoir, Richard Goodwin, the presidential speechwriter who authored that address, recalled hearing Johnson on the phone with King in the upstairs sitting room of the White House residence. King had watched the speech from the living room of the family of Jimmie Lee Jackson, a civil rights protester whose shooting death at the hands of an Alabama state trooper had helped inspire the Selma marches.

“Thank you, Reverend, but you’re the leader who’s making it all possible. I’m just following along trying to do what’s right,” Goodwin recounted the president telling King.

Alice Crites and Ann Hornaday contributed to this report.