By Alex Rogers and Manu Raju, CNN

(CNN) -- President Donald Trump indicated to associates during a flight on Air Force One on Wednesday that he would not intervene in the US Senate Republican primary in Kansas despite the fears among top Republicans that the state could elect a nominee who will lose the seat and thus the Senate, according to three sources with knowledge of the conversation.

While the GOP establishment has long been alarmed by the prospect that conservative firebrand Kris Kobach could win the primary on Tuesday only to lose the general election in November, Trump has so far not endorsed its favored candidate, Republican Rep. Roger Marshall. Trump has spoken with both Marshall and Kobach over the past several months but has never seemed highly motivated to make an endorsement, even when he's pushed by his close allies, according to a White House official.

That appeared to again be the case during the flight from Texas. One source said that Texas Sen. Ted Cruz chimed in during the conversation, reminding Trump that Marshall had initially supported then-Ohio Gov. John Kasich in the 2016 presidential race. Trump responded that he would not weigh in on Kansas' Senate primary.

Cruz told CNN on Thursday that he would not divulge details of a "private conversation with the President." The senator added, "I have not taken a position in that race."

While a Democrat has not won a US Senate seat in Kansas since 1932, the GOP primary race could very well determine which party controls the Senate next year. The nonpartisan Cook Political Report recently concluded that the Democrats are a "slight favorite" to take back the Senate as the President's popularity has plummeted.

Establishment Republicans are concerned that if Kobach wins the party's nomination, the race will be a repeat of 2018, when he lost the gubernatorial race to Democrat Laura Kelly. State Sen. Barbara Bollier, a former Republican running for the Democratic nomination, has broken state fundraising records by bringing in $7.8 million, according to the latest Federal Election Commission filings.

"Republican senators are very worried about this," said a Republican strategist granted anonymity to discuss a delicate matter involving the President. "This is what contributes to his lack of connection to Senate Republicans. They don't feel like he has their back."

Marshall represents a solidly Republican, farm-focused district from which former Senate Republican leader Bob Dole, Sen. Jerry Moran and the retiring Sen. Pat Roberts built the base of their power. Dole and Roberts have both endorsed Marshall.

Roberts told CNN that this was the first time that he has endorsed in a Republican primary. "I think the stakes are so high that it was the thing to do," said Roberts. He said "well, yeah" it would be helpful for the President to endorse Marshall but did not think Trump would do so.

"Hell, everybody can always do more," Roberts said when asked if Trump should say something that would be helpul to Marshall in the primary.

But Kobach, an anti-illegal immigration and voter-fraud crusader, has built a solid base of support in the state, gaining a passionate following as a former Kansas secretary of state and state party chairman. In 2017, he led Trump's voter integrity commission, which disbanded without finding any evidence of widespread voter fraud. In 2018, he won his party's nomination by only 343 votes over then-Gov. Jeff Colyer, and then lost to Kelly by 5 points. Kobach has since worked with We Build the Wall, a nonprofit advocacy group that claims to have raised $25 million to build a wall on the US-Mexico border.

Ronald Thacker, GOP chairman of Douglas County in the eastern part of the state, told CNN he supports Kobach because of their alignment on the issues: anti-abortion rights, anti-illegal immigration and in favor of stringent voter ID laws. But he acknowledged that he is "very concerned" Kobach would not do well in the state's major cities, where "even the Republicans are not very conservative."

A wild card in the race is the campaign of Bob Hamilton, who owns a plumbing company. It has spent more than $2.5 million in ads, according to Kantar's Campaign Media Analysis Group. In one of his latest, a man says that Hamilton is a "crazy, Trump-supporting, American-loving, flag-waving conservative." Hamilton is then shown water-skiing, holding the American flag.

"Kobach can't win," adds the man. "And Marshall? He's one hot mess, moderate squish."

Democrats have meddled in the Senate Republican primary race in order to elevate Kobach's campaign, which has raised less than $1 million, according to the latest FEC data. A Democratic-aligned group called the Sunflower State PAC has spent more than $4.6 million on anti-Marshall advertising, according to Kantar's Campaign Media Analysis Group, the most of any organization. One ad calls Marshall "part of the Washington swamp" who is beloved by "Mitt Romney Republicans" and Never Trumpers.

But a rival Republican-linked group, Plains PAC, has countered, spending more than $3.3 million on ads attacking Kobach, claiming he has "ties to White nationalists." (Kobach's campaign has rejected those ads as "false.") The Senate Leadership Fund, an organization aligned with Senate Majority Leader Mitch McConnell, and the US Chamber of Commerce have also spent a combined more than $2 million supporting Marshall, emphasizing the congressman's work on behalf of veterans.

"I think it's going to be helpful if the people of Kansas choose the most conservative candidate in their primary who can actually win a general election," Indiana Sen. Todd Young, the National Republican Senatorial Committee chairman, told CNN. "I do think they'll take into account what happened in the recent gubernatorial election."

Last month, Trump called David McIntosh, the head of the powerful anti-tax group Club for Growth, asking him to take down its ads attacking Marshall as being insufficiently conservative, according to The New York Times. McIntosh told CNN that Marshall is "not a strong pro-growth candidate" but the group would not endorse and its super PAC would deploy resources elsewhere.

But some Republicans would still like to see Trump send a stronger signal indicating his preference for the Senate nomination, believing that it could throw a close race to Marshall and make up for the President's mistake in endorsing Kobach during the 2018 gubernatorial primary.

Ryan Flickner, a top official at the Kansas Farm Bureau, which has endorsed Marshall, told CNN that a Trump endorsement could be crucial.

"Kansas Republican voters know an endorsement from President Trump helps protect the seat, and his endorsement of Marshall would be welcome news to help get our KFB-endorsed candidate across the finish line," said Flickner. "All across agriculture, President Trump's voice carries a lot of weight. Farmers and ranchers listen and want to know what he thinks."