Uber’s board plans to approach Facebook operating chief Sheryl Sandberg about filling its newly vacant CEO spot, The Post has learned.

The Sandberg idea is being pushed by Uber board member Arianna Huffington in the wake of Travis Kalanick’s surprise resignation as chief executive late Tuesday, according to a source close to the board.

Insiders say former Disney operating chief Tom Staggs is also a leading candidate for the CEO job. Indeed, Staggs already interviewed at the company ahead of Kalanick’s exit, according to one source, as the ride-sharing app scrambles to recover from a slew of scandals this spring.

But “Sheryl is their first choice,” according to one source briefed on the situation, adding, “That doesn’t mean they’re going to get her.”

Representatives for Uber and Sandberg didn’t immediately respond to requests for comment. A spokesperson for Huffington wasn’t immediately available to comment.

Other women said to be on Uber’s list include Marissa Mayer, who just stepped down as chief executive of Yahoo after selling the search pioneer to Verizon. CVS exec Helena Foulkes, who reportedly had been a candidate to fill the No. 2 spot at Uber before Kalanick’s resignation, is also in the mix.

AOL CEO Tim Armstrong has also been a rumored candidate, as has Turner CEO John Martin.

Nevertheless, sources say Uber directors, Huffington especially, are increasingly convinced that a woman would be ideally suited to fix Uber’s mess.

Last week, after former US Attorney General Eric Holder delivered a scathing report that blasted Uber’s culture for rampant sex harassment, a woman sued the company, alleging an executive had shared her medical records with top brass after an Uber driver raped her.

“Sheryl is seen as exactly what this company needs right now,” according to a source, noting Uber has gotten a reputation for being a cesspool for sex harassment. “It starts with the optics.”

Nevertheless, few think it’s likely Sandberg would move to such a post given her long history and success at Facebook — and despite her close friendship with Huffington.

Kalanick resigned as CEO under pressure from key shareholders including Benchmark Capital, Menlo Ventures and Fidelity, which delivered a letter demanding he step down on Tuesday, sources said. Still, insiders note that Kalanick is keeping his board seat at Uber and still has majority voting control over the closely held company’s shares.

Staggs, meanwhile, is currently without a portfolio and has interviewed for several positions including the Sony Entertainment CEO post, which wasn’t ultimately a fit for him.

At Uber, a new CEO would have fires to fight on several fronts, not least altering the bro culture that has grown at the firm to the disgust of female employees.

Kalanick has been a controversial figure for those who want to see Uber execute a public offering quickly and for those who advocate for a more corporate approach to the business.

Kalanick is currently facing both personal and professional calamity. His mother was killed in a boating accident and his father was injured.

Uber, which began as a technology to facilitate ride sharing in cars, is also getting into helicopters and boats and the delivery business.