Donald Trump and Mexican President Trade Barbs Over Money for Border Wall

— -- Following a cordial meeting with Mexican President Enrique Pena Nieto Wednesday that Republican presidential nominee Donald Trump characterized as a "substantive, direct and constructive exchange of ideas," the two men took the dispute over who would pay for Trump's border wall to Twitter.

At a press conference in Mexico City following his meeting with Pena Nieto, Trump said that the two had not discussed who would pay for the wall. He explained that the conversation would come "at a later date."

But later Wednesday evening, Pena Nieto tweeted that the topic led off their conversation and that he was unambiguous about his country’s stance.

"At the beginning of the conversation with Donald Trump I made it clear that Mexico will not pay for the wall," wrote Pena Nieto in Spanish.

Trump, meanwhile, flew to Arizona for a highly-publicized speech about his immigration reform policies and was direct in his own right about who he believed would be footing the bill.

"We will build a great wall along the southern border," said Trump. "And Mexico will pay for the wall."

He then repeated the sentiment on Twitter early Thursday morning, saying, "Mexico will pay for the wall!"

Pena Nieto replied Thursday afternoon, quoting Trump’s tweet and saying in Spanish, "I repeat what I told you personally, Mr. Trump: Mexico will never pay for a wall."

The Twitter exchange comes on the heels of an op-ed written by Pena Nieto in the El Universal newspaper Thursday morning, describing the rationale behind inviting Trump to Mexico.

In the essay, Pena Nieto explains he was "very clear" with Trump that Mexicans felt offended and hurt by his negative declarations about their country and again stated that they would not pay for a wall.

Trump appeared to be undeterred by any disagreement with Pena Nieto prior to his speech in Arizona, tweeting upon his return to the U.S. that Mexico had "wonderful leadership and high quality people."

"[I] look forward to our next meeting," he said.

ABC News' Katherine Faulders contributed to this report.