Bette Davis reportedly once said of her Hollywood arch nemesis, “I wouldn’t piss on Joan Crawford if she were on fire.”

But while the new FX miniseries “Feud: Bette and Joan” — premiering March 5 with Jessica Lange as Crawford and Susan Sarandon as Davis— focuses on the rivalry that brewed between the Hollywood legends while filming 1962’s “What Ever Happened to Baby Jane?,” their animus actually began decades before.

It was fueled by competition over movie parts, Academy Awards and even a man: Franchot Tone, Davis’s co-star in 1935’s “Dangerous.” Davis was said to have fallen for him, but he was already dating Crawford. In the dishy book “Bette and Joan: The Divine Feud,” Shaun Considine writes that Tone loved working with Davis, prompting Crawford to wed him shortly after “Dangerous” wrapped.

“[Tone] picked the more exciting, superficially sexier woman,” Ben Mankiewicz, a host on Turner Classic Movies, told The Post.

In 1987, Davis said in an interview with journalist Michael Thornton: “[Crawford] took him from me. She did it coldly, deliberately and with complete ruthlessness.”

Davis won the Best Actress Oscar for “Dangerous,” but Crawford could not bear Tone paying any attention to his co-star at the ceremony. Considine writes that Tone “leapt to his feet as Bette passed by and embraced her.” Crawford, meanwhile, gave Davis’ simple dinner dress the once-over and said with dripping sarcasm, “Dear Bette! What a lovely frock!”

Still, in 1943, when Crawford left MGM for Warner Bros. — the studio home of Davis — she sent her enemy flowers. “She wanted [Davis’] approval, but Bette brushed her aside. She had no patience with her,” said Sandford Dody, ghostwriter on Davis’ 1962 autobiography “The Lonely Life.”

‘It was fueled by competition over movie parts, Academy Awards and even a man: Franchot Tone, Davis’s co-star in 1935’s ‘Dangerous.”

Davis felt glamour girl Crawford was nowhere near her league when it came to being a serious actress. “Director Vincent Sherman said, ‘To get Joan to cry you had to tell her a sad story,’” said Ed Sikov, author of “Dark Victory: The Life of Bette Davis.” “Bette just did it. That says a lot about their natures as actresses.”

Oscars would figure prominently in the women’s conflict, beginning in 1945 when Davis turned down the starring role in “Mildred Pierce.” Scooping up Davis’ sloppy seconds, Crawford won her only Academy Award. Seven years later, Davis filmed “The Star,” which would lead to one of her 11 Oscar noms. The story of a washed-up actress desperate to reignite her career, it was written by a former pal of Crawford’s — and presumed to be about the actress. (Davis told Playboy in 1983, “Oh, yes, that was Crawford.”)

But by 1961, the studio system that had nurtured the actresses through World War II — the era of “women’s pictures” — had collapsed. Both stars, now middle-aged, “needed a hit,” said Mankiewicz, whose great-uncle Joseph directed Davis in “All About Eve.”

“On top of that,” he added, “they both needed the money.”

Their unlikely rescue came via the horror film “What Ever Happened to Baby Jane?” — about a demented former child star (Davis) who holds her crippled sister (Crawford) captive. Crawford signed on first, but knew she needed the perfect co-star to make it a career-reviving hit. Putting ego aside, she flew to New York and asked her rival to take the part.

Amazingly, Davis did.

Ryan Murphy, who developed “Feud,” finds it incredible that Crawford would be willing to ask — and that Davis, with whom he struck up a fan-letter correspondence in the 1970s — accepted. “It makes no sense that [Crawford] would go to Bette Davis with the ‘Baby Jane’ script. For years, she claimed that Bette tried to steal . . . Franchot Tone. It was odd that she felt Bette was the only person for that part.”

As seen in “Feud,” studio patriarch Jack Warner (Stanley Tucci) initially refused to finance the film, saying, “I wouldn’t give you a dime for those two washed-up old broads.” The women prevailed, but trouble started even before they reported to the set.

At a press conference for their contract signings, Davis snagged the best spot at the table — ensuring that her name would appear first in photo captions. Crawford quickly stood to Davis’ left. Now, her name would be first.

“Feud” re-creates this scene — and how Crawford was devastated to see that Davis’ contract was worth more than hers. “In addition to Bette’s $60,000 salary, [Davis] was to receive $600 per week living expenses,” Considine writes.

Davis in particular had a knack for using a subtle, sly touch when twising the knife. She went so far as to have a Coca-Cola machine installed in her dressing room — because Crawford had been married to Pepsi CEO Alfred Steele.

While filming, Davis spontaneously kicked her co-star in the head during a fight scene. Crawford got her revenge: Knowing Davis had a bad back, she wore a lead-lined weight-lifter’s belt under her costume for a scene in which Davis had to drag her across the floor.

‘Baby Jane” was a big hit, and Davis — much to Crawford’s fury — was nominated for yet another Best Actress Oscar. Crawford went on the offensive, calling Davis’ fellow nominees Anne Bancroft and Geraldine Page, both in New York, to offer to accept the Oscar for them if they couldn’t attend the ceremony.

Davis was favored to win — but Bancroft’s name was called and Crawford grandly accepted. Davis told the LA Times, “I almost dropped dead when I didn’t win.” In her 1987 memoir “This ’N That,” she elaborated. “Moments later, Crawford floated down the hall, past my door. I will never forget the look she gave me. The look clearly said, ‘You didn’t win, and I am elated!’ ”

“The rule of thumb was that an Oscar winner usually added at least a million dollars to the box office receipts of a film,” Davis later said. “Since Joan and I each had a percentage of the movie, how Medean, how foolish she was to work against my winning.”

The studios wanted more, of course, and the women signed on for 1964’s “Hush . . . Hush, Sweet Charlotte.” Crawford left less than two weeks into filming, claiming illness; most agreed she simply didn’t want to be upstaged by Davis again.

“Joan knew she didn’t have the acting chops [to do the part],” said author Sikov. The part was recast with Olivia de Havilland.

When Crawford passed away in 1977, the feud did not die with her. Davis allegedly said, “You should never say bad things about the dead, only good . . . Joan Crawford is dead. Good.”

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