Allies of President Trump are pursuing an effort to acquire right-leaning news channel One America News Network, according to people familiar with the matter, in a bid to shake up a conservative media market that has been dominated by Fox News.

The investment firm Hicks Equity Partners is looking to acquire the channel and is pitching other wealthy GOP donors to arrange a bid of roughly $250 million for the channel’s parent company, the people said. The firm is owned by the family of Thomas Hicks Jr., co-chairman of the Republican National Committee and a close friend of Donald Trump Jr.

The efforts come as Mr. Trump has periodically rebuked Fox News for being too critical—despite its opinion-show hosts’ general support of his administration—and has praised One America News Network. The channel’s opinion programming is known among its cable-news peers for its praise of Donald Trump and its advocacy for conservative causes.

“Over the last 25 years, we have been active media investors. Any interest in OANN is from a purely business standpoint,” said Rick Newman, a partner at Hicks Equity Partners who is leading the deal with family patriarch and Trump donor Thomas Hicks.

One America News Network President Charles Herring said in an email that the network has become “a reliable news source for a national audience,” adding that the company has seen interest from potential suitors from time to time. “With the 2020 political season in full swing, expressed interest is on the rise,” he wrote. “Yet our family didn’t build our operations to sell it.”

One America News is owned by San Diego-based Herring Networks Inc., a TV-programming company that was launched in 2004 by Mr. Herring’s father, Robert Herring Sr., founder of a circuit-board firm. He launched the news channel in 2013.

Share Your Thoughts Do you think the conservative media market could use more competition? How would you vet a new network? Join the conversation below.

The network draws a fraction of the viewers of the cable news industry’s titans—including ratings leader Fox News, MSNBC and CNN—and isn’t as widely distributed.

Last year, One America News said it was available in 35 million homes. Prospective investors have been told a new owner could increase the channel’s reach by striking carriage deals with additional pay-TV companies.

The Hicks firm would bring in new management, upgrade production values and offer coverage that caters to viewers whose political outlook is in the center and right-of-center, segments it views as underserved, the people familiar with the matter said.

Fox News parent Fox Corp. FOX -0.97% and Wall Street Journal parent News Corp NWS 1.85% share common ownership.

People close to Mr. Hicks Jr. said the RNC co-chair isn’t involved in efforts to buy OANN and had resigned his position as a partner at Hicks Holdings to avoid possible conflicts of interest after taking the RNC post. He and other family members could decide to opt into the deal, one of the people said. Another person said Mr. Hicks Jr. was unlikely to participate while he is an official at the RNC.

Mr. Hicks Jr. didn’t respond to requests to comment.

Hicks Equity Partners is a subsidiary of Hicks Holdings, the Dallas family office controlled by the elder Thomas Hicks that invests the wealth of the Hicks family. It is in talks with roughly a dozen wealthy Republican donors about the deal.

Doug Deason, a Dallas investor who has donated to the Trump campaign, said in an interview that he is planning to join the bid. Elliott Management Corp., the hedge fund run by Republican megadonor Paul Singer, was pitched but wasn’t interested, said people familiar with the matter.

The Hicks firm also plans to pitch Todd Ricketts, the RNC finance chairman, according to a person familiar with the matter. “The Ricketts have great respect for the Hicks family and always keep an open mind for good ideas,” said Brian Baker, a spokesman for the Ricketts family.

Part of the discussion among investors about buying the network has centered on the desire to strengthen the conservative voice in media coverage, which they feel offers far more options for liberal audiences, Mr. Deason said.

“The concern is that there’s only Fox. The left has so much more,” Mr. Deason said.

Some Republican donors say privately that Fox News isn’t doing enough to toe the party line, according to people familiar with the matter. They have noted, for instance, that the network doesn’t always carry the president’s full campaign rallies live on air, the people added.

A Fox News spokeswoman didn’t immediately respond to a request for comment.

One America News Network, also known as OANN, has served as a launchpad for prominent conservative commentators, such as current Fox Nation host Tomi Lahren.

Mr. Trump has favorably contrasted the smaller cable channel with Fox News.

Though the president routinely cites Fox News’s coverage and quotes its programs, he also has expressed some dissatisfaction with the network in recent months. “Only pro Trump Fox shows do well. Rest are nothing,” he tweeted in December.

Last year, before Fox News anchor Shepard Smith left the channel in August, he tweeted, “Watching Fake News CNN is better than watching Shepard Smith, the lowest rated show on @FoxNews. Actually, whenever possible, I turn to @OANN!”

In December, One America News worked with Mr. Trump’s personal lawyer, Rudy Giuliani, on a series of interviews with current and former Ukrainian officials aimed at justifying Mr. Giuliani’s push for investigations in Ukraine that could benefit the president politically. The attempt to press Ukraine to launch those investigations helped set off the impeachment inquiry of the president the fall. The president has denied any wrongdoing.

Mr. Trump retweeted several posts by the network about the series and that month praised One America News for “doing incredible reporting.”

The Hicks family’s wealth stems from the senior Mr. Hicks’s co-founding of the now-defunct private-equity firm Hicks Muse Tate & Furst Inc., a leading media investor of the 1990s and 2000s. Its portfolio included hundreds of radio stations that made up what was then the largest radio broadcasting company in the country. Hicks Muse sold that company to Clear Channel Communications Inc., now known as iHeartMedia Inc.

Hicks Equity Partners hasn’t viewed the landscape for media deals as attractive in recent years, believing valuations were inflated and the industry was fully consolidated, said a person familiar with the firm’s views.

The family is also known for investing in sports teams. The family at one time owned teams including the English soccer club Liverpool, the Texas Rangers baseball team and the Dallas Stars hockey team.

—Benjamin Mullin contributed to this article.

Write to Juliet Chung at juliet.chung@wsj.com, Corrie Driebusch at corrie.driebusch@wsj.com and Rebecca Ballhaus at Rebecca.Ballhaus@wsj.com