Liz Cheney has released her concession call to Harriet Hageman, after the Trump-backed Republican who won the Wyoming US House primary on Tuesday told Fox News her rival had not made “any kind of concession or anything else”.

Cheney gave a recording of the call to Politico. In it, she said: “Hi, Harriet, it is Liz Cheney calling. It is about 8.13 [pm] on Tuesday the 16th. I’m calling to concede the election and congratulate you on the win. Thanks.”

Hageman beat Cheney, a three-term congresswoman, in a landslide.

Cheney’s work as vice-chair of the House January 6 committee, and opposition to Donald Trump, sealed her fate in Wyoming, a deep-red state once represented by her father, the former vice-president Dick Cheney.

In her concession speech, delivered in Jackson against the backdrop of a setting sun and the Teton mountains, Cheney outlined plans to oppose Trump from outside the House.

In her speech, she cited the Republican president Abraham Lincoln, calling him “the great and original champion of our party” and noting that the 16th president was defeated in elections for the Senate and the House before he won the most important election of all.

“Lincoln ultimately prevailed,” Cheney said. “He saved our Union and he defined our obligation as Americans for all of history.”

She also mentioned Ulysses S Grant, who was a general before becoming the 18th president: “Lincoln and Grant and all who fought in our nation’s tragic civil war, including my own great-great-grandfathers, saved our union. Their courage saved freedom. And if we listen closely, they are speaking to us down the generations. We must not idly squander what so many have fought and died for.”

On Wednesday morning, Cheney told NBC she was considering a White House run. She has formed a fundraising committee called The Great Task, a nod to the Gettysburg Address, which Lincoln delivered in 1863.

Most observers, however, say Cheney does not have a path to primary victory in a Republican party dominated by Trump and his followers. As a strong conservative, she seems unlikely to attract significant Democratic or independent support.

Hageman, an attorney, once opposed Trump herself but came round to win his endorsement. She has repeated Trump’s lie about the 2020 election, which Trump lost conclusively to Joe Biden, being rigged.

She will be heavily favoured to win Wyoming’s sole seat in the US House in November, against the Democrat Lynette Grey Bull.

Earlier on Wednesday night, Hageman discussed the concession call with the Fox News host Sean Hannity.

She said: “I haven’t had any other contact with Liz Cheney. She made the one effort and all she said was ‘Hello, Harriet.’ And then that was the end of it.”

Hageman repeated her claim that Cheney did not concede and her spokesperson provided Politico with a video of a phone playing a message apparently from Cheney. The website reported: “The message as played on-screen contains only Cheney’s two-word greeting before sound cuts off for more than 10 more seconds.”

Cheney told Politico she called Hageman three times, leaving the voicemail before going out to speak.