Donald Trump has dismissed the Mexican president's warning that his country will not pay for a wall along the US border to clamp down on illegal immigration.

"They don't know it yet, but they're going to pay for it," he said just hours after making a surprise visit to Mexico to meet with President Enrique Peña Nieto.

The Republican candidate was in bombastic form back on home soil in Arizona as he unveiled a hard-hitting plan to curb illegal immigration from its southern neighbour.

He told cheering crowds of his plan to fit the wall along the America's Mexican border with sensors to detect tunnels used for trafficking people and drugs - vowing to significantly increase patrol staff along the border.

His 10-point plan includes an end to "catch and release" - the practice of letting illegal immigrants go immediately after they are detained. 

Instead, Mr Trump pledged to "take them great distances" in their home country to deter them from attempting to cross into the US again.

The candidate also devoted considerable time in his landmark speech to the millions of illegal immigrants already in the US.

Mr Trump poured scorn on Barack Obama's amnesty for law-abiding illegal immigrants who live and work in the US, and said those who have overstayed their visas will be returned back to their home country and made to re-apply for citizenship.

A biometric visa system would also be reintroduced to end visa overstays for good - and he said two of the attackers on 9/11 could have been thwarted if such a system had been introduced sooner.

He pledged there would be a "zero tolerance" for migrants who commit crime in the US - and he would force their home countries to take them back, even if they didn't want to.

Criminals who attempted to re-enter the US would then be given "strong" sentences.

Mr Trump also proposed "ideological certification" of immigrants to the US "to make sure that those we are admitting to our country share our values and love our people". 

Visas for countries where adequate screening cannot occur would also be suspended, he said.

He warned it was dangerous to offer an open door to refugees from the likes of Syria and Libya where there was no paperwork and little detail on their background. 

Even though Mr Trump and Mexico's president appeared to end their meeting on good terms, Mr Peña Nieto later said some of the policies detailed in the Republican's speech represented a threat to Mexico.

He tweeted last night that he had made it clear to Mr Trump that Mexico will not pay for the proposed wall.