He also signed an action to strip funds from US cities that are sanctuaries for undocumented immigrants.

Mr Trump said in a TV interview that Mexico would “absolutely, 100%” reimburse the US for his wall.

Building a 2,000-mile barrier along the Mexican border was one of his key pledges in the election campaign.

Mr Trump spoke of a “crisis” on the southern US border as he signed the actions during a ceremony at the Department of Homeland Security on Wednesday.

“A nation without borders is not a nation,” he said. “Beginning today the United States gets back control of its borders.”

Mexicans see wall as racist – Will Grant, BBC News, Mexico City

Mexicans might disagree about a lot – especially on the political and economic direction of the country.

But if there is one thing around which almost all of Mexico can coalesce, it is their profound opposition to the US border wall.

From the Mexican side of the border, it is seen as a policy which is intended to break up families and prevent ordinary people from looking for seasonal work in the north – the kinds of jobs, they note, which prop up the US economy.

Others go further, and consider the entire border wall to be a racist and xenophobic policy.

Either way, Mexicans from the president’s office to the factory floor agree that the country will not pay for a wall they don’t want and didn’t call for.

They say they won’t finance the project either at the time of building or in the future.

That’s not to say some Mexicans aren’t in favour of change in terms of bilateral immigration.

Many are aware of the risks that their countrymen take by crossing illegally into the US, especially through dangerous border regions such as the Arizona desert.

Rather than a wall with their neighbour to the north though, they want to see comprehensive immigration reform including guest worker programmes and temporary work visas.

Mr Trump said relations with Mexico – whose President Enrique Pena Nieto he is scheduled to meet at the end of the month – would get “better”.

The directives are among several expected on national and border security this week.

He is next expected to announce immigration restrictions from seven African and Middle Eastern countries, including Syria, Yemen, and Iraq.

Mr Trump is also tipped to halt access to the country for some refugees – until the vetting process can be made more rigorous.

Earlier on Wednesday, Mr Trump told ABC News he would recoup costs of the wall – for which the US Congress would need to approve funding – from Mexico.

“There will be a payment. It will be in a form, perhaps a complicated form,” he said.

“It’s also going to be good for Mexico”, he added.

Mr Trump has previously estimated the wall would cost $8bn (£6.4bn), but critics have suggested it could be nearly double that sum.

Earlier on Wednesday, Mr Trump also promised a “major investigation into voter fraud”.

He tweeted that the inquiry would focus on illegal votes in two unspecified states and dead registered voters.

This week he claimed that between three and five million illegal immigrants had voted for Hillary Clinton, but offered no evidence. – BBC