President-elect Donald Trump on Sunday chose Republican National Committee Chairman Reince Priebus as his chief of staff and named controversial, “alt-right” media exec Steve Bannon his chief strategist and senior counselor.

The announcement followed widespread reports that both men were in the running to be chief of staff — and repeatedly gave Bannon top billing regarding their respective roles when Trump takes office in January.

“Bannon and Priebus will continue the effective leadership team they formed during the campaign, working as equal partners to transform the federal government, making it much more efficient, effective and productive,” the statement from Trump’s transition team said.

US Rep. Peter King (R-LI) called the appointments “basically two sides of the same coin.

“It’s showing what Donald Trump’s style is going to be,” King said.

“Having Bannon as strategist shows he’s certainly committed to conservative principles but by having Reince Priebus, he’s showing he’s having someone who is conservative but who has the insider experience to get things done.”

But Bannon’s reputation as a right-wing firebrand whose Breitbart News website has been accused of racism, anti-Semitism and Islamophobia led to alarm in some corners of the Republican establishment.

“The racist, fascist extreme right is represented footsteps from the Oval Office. Be very vigilant America,” tweeted John Weaver, a strategist for Republican Ohio Gov. John Kasich, who lost to Trump in the primaries.

Meanwhile, GOP political consultant Roger Stone, a longtime Trump insider, warned in a tweet before the announcement that making Priebus chief of staff “would cause a rebellion in Trump’s base.”

Stone added the hashtag #RyansBoy in apparent reference to House Speaker Paul Ryan, who, like Priebus, is from Wisconsin. Priebus remained neutral during the combative GOP primaries but embraced Trump when he emerged the victor, even as leading party figures — including 2012 presidential nominee Mitt Romney — publicly bucked his candidacy.

Bannon was named CEO of Trump’s campaign during an August staff shakeup. Following his appointment, The Post revealed that he had been accused of domestic violence by his second wife, and it was also reported that she accused him of making anti-Semitic remarks.

Trump said of the two key appointments: “I am thrilled to have my very successful team continue with me in leading our country.”

“Steve and Reince are highly qualified leaders who worked well together on our campaign and led us to a historic victory. Now I will have them both with me in the White House as we work to make America great again,” Trump added.

Priebus, 44, is a lawyer and longtime Republican operative who’s in his third term as national GOP chairman.

He once joked that his first name, which rhymes with “pints,’’ is a “cultural disaster,’’ the result of German and Greek parents. It is short for his real name, Reinhold.

Bannon, 62, is a former US Navy officer who worked for Goldman Sachs and was an early investor in the TV show “Seinfeld” before becoming a conservative filmmaker and taking over Breitbart following the sudden death of founder Andrew Breitbart in 2012.

Hours before Sunday’s announcement, Ryan — a frequent Breitbart target — raised no objections to Bannon becoming a major element of the Trump administration.

“I’ve never met the guy. I don’t know Steve Bannon so I have no concerns. I trust Donald’s judgment. I think he’s going to pick who he thinks best serves him and I’m sure we will work well with whoever is his chief of staff is, whoever his staff is,” Ryan told CNN’s “State of the Union.”