WASHINGTON — Republicans in Congress are beginning to grapple with how to help President-elect Donald Trump fulfill one of his biggest campaign promises: to build a wall along the Mexican border to slow illegal immigration.

Trump is expected to ask Congress to provide the initial funding for the project, estimated to cost from $12 billion to $38 billion.

Once construction gets underway, Trump has said he will demand reimbursement from the Mexican government, even though Mexican officials have said they will refuse to cooperate.

Amid concerns American taxpayers would shoulder the burden, Trump promised Friday that he would force the U.S. neighbor to pony up the money.

“The dishonest media does not report that any money spent on building the Great Wall (for sake of speed), will be paid back by Mexico later!” he tweeted.

Former Mexican President Vicente Fox responded on Twitter, “Trump may ask whoever he wants, but still neither myself nor Mexico are going to pay for his racist monument. Another promise he can't keep.”

Republicans, meanwhile, were scrambling Friday to figure out how to provide initial support for the wall, despite opposition from Democrats and resistance from budget conservatives in their own party.

“I'm not sure anyone believed the government of Mexico was going to write a giant check to the federal registry, but there are other ways to ensure Mexico does pay for the border buildout,” said Rep. Luke Messer, R-Ind., chairman of the Republican Policy Committee. “There are broad conversations about fulfilling all of Mr. Trump's campaign promises.”

An aide to House Speaker Paul Ryan said no new talks are underway with the Trump transition team on the border issue.

But Trump's transition team acknowledged that it would seek help from lawmakers.

“Obviously a centerpiece of Donald Trump's successful campaign was, ‘I'm going to build a wall and have Mexico pay for it,'?” adviser Kellyanne Conway said on “Fox and Friends.” “That hasn't changed, but Congress is examining ways to have the wall paid for through their auspices and Mr. Trump is making the point that he will have Mexico pay it back.”

During the campaign, Trump suggested that if Mexico did not agree to pay for the wall, the U.S. might impose a fee on financial remittances sent home by Mexicans working in the U.S. illegally. Such a move would surely provoke opposition from the financial services industry.

Messer said another possibility would be raising some of the funds, about $4 billion, by cutting a child tax break for immigrants here illegally.

The president-elect has also suggested the U.S. could try to pressure Mexico by reducing or slowing down the process by which Mexicans get travel cards and visitors' visas.

The U.S. Customs and Border Protection, parent for the Border Patrol, has already budgeted $175 million for “procurement, construction and improvements.”

But even if that money is diverted to the wall, it wouldn't be nearly enough.

On Capitol Hill, Republicans noted that President George W. Bush signed a 2006 law that authorizes construction of a wall along the nearly 2,000-mile border with Mexico. It was passed by a Republican-led Congress, with support from now Senate Minority Leader Charles Schumer of New York and other Democrats.

But it did not provide the needed funding.

Trump has estimated a wall would cost up to $12 billion to build. But it could be more than triple that, $38 billion, according to an analysis published by MIT Technology Review.

The difference depends on the type of the fencing — double versus single layer, for example — and how much of the border the wall covers.

Republicans said one scenario would be to link new border funds to a must-pass bill to keep the government running past April 28.

Such a legislative maneuver would force Democrats, who mostly oppose border funding absent broader immigration law changes, to join the vote or run the risk of a springtime government shutdown.

“They would have a hard time going home and explaining why they voted against it,” said one Republican leadership aide, granted anonymity to discuss private deliberations.

But the strategy could backfire if Republican deficit hawks raise concerns about adding to the debt load.

lisa.mascaro@latimes.com