The bombastic billionaire has promised to crack down on illegal immigration and insulted Mexican immigrants by calling them rapists, criminals and drug dealers.
As his party's presumptive presidential candidate, Trump is now eyeing a clash in the November election with his Democratic rival Hillary Clinton.
"There is no way to have Mexico pay (for) the wall. But any decisions inside (the) USA -- is a decision of its government," Pena Nieto said on Sunday.
The Mexican president earlier had said Trump's campaign pledge was a non-starter, but he did not address the issue as specifically. Pena Nieto said US-Mexican relations were based on coordination, collaboration and cooperation on security issues.
The Mexican president earlier compared Trump's rhetoric to the rise of European dictators Adolf Hitler and Benito Mussolini.