The GOP nominee told a cheering crowd that he would secure the border, and left open the possibility that millions of illegal immigrants be deported.
Hours earlier, Donald Trump met Mexican President Enrique Pena Nieto but said they had not discussed financing the wall.
President Pena Nieto later insisted he had told Donald Trump Mexico would not pay.
There had been speculation that Donald Trump would back off his plan to deport the estimated 11 million undocumented immigrants living in the US.
In his speech in Phoenix, there were conflicting signals about this. The Republican said their fate was not a “core issue” and that deporting “criminal aliens” would be the priority.
“We will treat everyone living or residing in our country with great dignity,” he said.
Later Donald Trump struck a more uncompromising note when he added: “Anyone who has entered the United States illegally is subject to deportation. That is what it means to have laws.”
He said it was the right of the US to choose immigrants that “we think are the likeliest to thrive and flourish and love us”.
Elaborating on that idea, Donald Trump said his “extreme vetting” would involve an ideological test for immigrants applying to live in the US.
“Applicants will be asked for their views about honor killings, about respect for women and gays and minorities, attitudes on radical Islam,” he said.
Donald Trump stormed to an unlikely victory in the Republican primaries partly due to his tough talking on immigration.
In Phoenix he vowed to protect the interests of Americans who he said lose out to new arrivals: “We have to listen to the concerns that working people, our forgotten working people, have over the record pace of immigration and its impact on their jobs, wages, housing, schools, tax bills and general living conditions.”
Donald Trump accused his Democratic rival Hillary Clinton of wanting to grant amnesty to undocumented immigrants and of advocating “open border” policies.
Reacting to Donald Trump’s visit to Mexico, Hillary Clinton said he had “choked” by not asking his hosts to pay for his wall.
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