They aren’t calling him “Lyin’ Ted” anymore.

Sen. Ted Cruz’s stock is rising among members of President Trump’s base who feel double-crossed after the president offered amnesty to illegals in the immigration debate. Some who opposed Mr. Cruz in his 2016 run for the White House are even clamoring for the Texas Republican to mount a primary challenge to Mr. Trump in 2020.

The newfound esteem for Mr. Cruz, whom Mr. Trump dubbed “Lyin’ Ted” when they battled for the 2016 Republican nomination, is coming from people who were die-hard supporters of the president.

“Ted Cruz has kept his word to the American citizens, and we are watching this very, very carefully,” said Sue Payne, a conservative activist in Washington’s Maryland suburbs who in 2016 championed Mr. Trump and reviled Mr. Cruz. “I’m getting very sick of Donald Trump right now. He’s turning his back on the people who put him in office.”

The moves by Mr. Trump that have alienated supporters include the budget deal that threatens more skyrocketing debt and a tax cut law that could make people pay more in high-tax states.

For many, however, the biggest double-cross was the offer of an immigration deal with a path to citizenship for 1.8 million Dreamers who came illegally to the U.S. as minors.

Mr. Cruz distinguished himself on that front last week by casting the only vote against opening the Senate immigration debate in which all three competing proposals included amnesty for Dreamers.

The effort collapsed when all three immigration plans failed to garner the 60 votes needed to survive. Mr. Cruz voted against every plan.

The president’s framework fell by a 39-60 vote, with 14 Republicans including Mr. Cruz opposing it.

William Gheen, president of Americans for Legal Immigration PAC, accused Mr. Trump of “dereliction of duty” by abandoning the get-tough policies that put him in the White House.

“This betrayal destroys his credibility and opens up a huge window of opportunity for Sen. Cruz or another GOP presidential primary challenger in 2020,” said Mr. Gheen, whose group advocates zero tolerance for illegals.

Mr. Cruz insisted that he is focused only on his run for re-election to the Senate in November, though he didn’t rule out another campaign for the White House down the road.

“My focus in the Senate has been to not worry about the political circus that is Washington and instead to focus on substance,” Mr. Cruz told The Washington Times.

For the past year, he said, the top four priorities for Republican majorities in Congress and the Trump administration have rightly been tax reform, regulatory reform, Obamacare repeal and judicial confirmations. He said the amnesty debate, as he termed it, was a serious mistake.

“I do not believe we should be granting a pathway to citizenship to anybody here illegally. I cannot for the life of me understand why so many Republicans are galloping to the left of Barack Obama on immigration,” he said. “Granting citizenship to people here illegally is inconsistent with the promises we made to the men and women who elected us.”

Shrugging off the Trump supporters moving in his direction, Mr. Cruz said, “My focus is on substance. The politics will take care of itself.”

Still, some of Mr. Cruz’s Republican colleagues were cheering the party’s embrace of amnesty for the Dreamers, who now face a March 5 phaseout of the Obama-era temporary protection from deportation known as DACA.

“The Republican Party has moved a long way — 1.8 million Dream Act-eligible people on a path to citizenship. That is a big change for our party,” said Sen. Lindsey Graham, a South Carolina Republican who spearheaded a bipartisan push for amnesty measures.

Kevin Madden, a Republican Party strategist who was a senior adviser to Mitt Romney’s 2012 presidential campaign, discourages Mr. Cruz from becoming too confident after hearing chatter about his presidential potential.

Mr. Trump might have tested his base on immigration, but he hasn’t lost it. The president’s approval rating among Republican voters is above 80 percent, he said.

“Any primary challenge is going to be very difficult,” Mr. Madden said.

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