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President Trump cut a last-minute robocall for Republican Ron Estes in the special election to fill Kansas' empty seat in Congress - raising the stakes in the race and fueling speculation over whether the contest has become too close for comfort for the GOP.

The Trump call coincided with a visit Monday from Sen. Ted Cruz of Texas. It also followed a pro-Estes robocall from Vice President Mike Pence and recent investments from the National Republican Campaign Committee, as part of a late push to beat back Democrat James Thompson.

"Hello. This is President Donald Trump and I have something big to tell you," Mr. Trump said in the call. "On Tuesday, Republican Ron Estes needs your vote and needs it badly."

The president said Mr. Estes "is a conservative leader who is going to work with me to make America great again. Ron is going to be helping us, big league."

"But I need Republicans like Ron Estes to help me get the job done," Mr. Trump said.

Democrats, meanwhile, are hoping a last-minute liberal surge driven by antipathy to Mr. Trump and Kansas Gov. Sam Brownback will lead to an upset in Tuesday's special election to fill left empty by former Rep. Mike Pompeo, who became the new CIA director.

Political observers say district demographics and recent history suggests that Mr. Estes is well-positioned to win, but say Mr. Brownback's ongoing fights with his own party and Mr. Trump's early stumbles as president have left a restless electorate.

"If you look how Pompeo had won re-election numerous times and the percentages he won by this should be a good, solid, easy victory for Estes," said Matt Hickam, a Kansas-based GOP strategist.

"But I think the thing to watch, and this might be a thing to watch in other districts in the country, is how motivated, how amped up, are [Sen. Bernard] Sanders supporters and the left wing of the Democratic Party," Mr. Hickam said. "Are they able to turnout voters because of a hatred of Trump and conservatives?"

Burdett A. Loomis, a political science professor at Kansas University, said that in a normal year Mr. Estes wins, but predicted it will be a low turnout affair and that Democrats will have more enthusiasm.

"He's an unexciting candidate," Mr. Loomis said, alluding to Mr. Estes. "Honestly, he's a stiff."

Mr. Trump tapped three House lawmakers for high roles in his administration, creating a series of open seats that are serving as early political tests. The seats are all in solidly red states, but Democrats are still hoping for an upset.

The marquee race is next week's showdown in Georgia, where Democrat Jon Ossoff is touting his anti-Trump credentials and leading the crowded field in the race to replace former Rep. Tom Price, who was tapped to be Mr. Trump's health secretary.

Mr. Trump's phone call, though, showed that Kansas is heating up as Republicans look to bolster Mr. Estes in his race against Mr. Thompson. A loss in GOP country would be embarrassing.

The NRCC has reportedly invested almost $100,000 into the contest.

"I am not sure that is a sign of Estes' possible weakness or if it is just the Republican leadership just wanting to make sure this thing is tied up with a bow," Mr. Hickam said.

Mr. Trump carried the district in south central Kansas by nearly 30 percentage points over Hillary Clinton in the 2016 presidential race. The district includes Wichita, which has been a front in the culture wars over abortion for decades.

Mr. Thompson is pro-choice. Mr. Estes is pro-life, which has helped him rally local activists to his campaign in the district which houses the national headquarters of Operation Rescue and its clinic-blockading efforts that go back to the 1990s in Wichita.

The NRCC is running ads claiming that Mr. Thompson supports late-term abortions and taxpayer-funded abortions, and touting that Mr. Estes "defends the culture of life."

The Thompson campaign says the accusations are false, and is trying to tie Mr. Estes to Mr. Brownback, who late last month vetoed a measure that would have expanded Medicaid under Obamacare.

"Ron Estes is nothing but a Brownback yes-man," Thompson spokesman Chris Pumpelly said in a recent fundraising email. "Kansas just can't afford another Brownbacker."

Democrats relied on a similar strategy in legislative races in the fall and picked up seats in both the Kansas House and Senate.

"All in all, a Thompson victory would be a huge upset," Mr. Loomis said. "Unlikely in this partisan era, but if D's are energized and turn out, who knows. Clearly R's in D.C. are worried."

David Boyer contributed to this article.