President Donald Trump said Wednesday that construction of his promised U.S. -Mexico border wall will begin "in months". A White House official did not immediately comment on the discrepancy.
Mr Trump signed an executive order calling for a "large physical barrier" between the U.S. and Mexico.
The spat escalated this week as Trump ordered the wall's construction, which he wants Mexico to fund, while Pena Nieto cancelled a meeting with the U.S. president scheduled for next week in Washington.
In response, Trump tweeted on Thursday that if Mexico is unwilling to pay for the cost of the wall, that he will cancel the upcoming meeting planned for January 31.
Mr Trump said the call was friendly and he looked forward to renegotiating the USA trade relationship with Mexico in the future.
"We don't want to have an open border, but I do know that if you put an emphasis on a 14th century solution called a "wall" - we saw what happened with Imperial China and their wall, we know what happened with the Communists and their wall".
Trump's executive order last week to start building the wall - fulfilling one of his campaign promises - has already stirred controversy, leading to a diplomatic crisis with Mexico. "Through the wall, not only do we secure our border, but I think we're going to save additional money that we would have spent tracking down illegal immigrants". Trump now says American taxpayers will foot the bill initially, but that we will be "reimbursed" by Mexico. In the meantime, US taxpayer dollars will be put toward construction of the wall.
We understand the emotional appeal of building a wall to solve hard problems in the USA immigration system and national security - just build a wall to keep the bad people out, problem solved! - but we must deal in facts. During the visit "to honor and remember the victims of the Holocaust, Foreign Secretary Dr. Luis Videgaray Caso expressed the affection of Mexico towards the people of Israel".
In addition, during an interview with American channel Fox News on Thursday, Trump justified his decision, citing Israel which, he said, had managed to "stop 99.9 percent" of illegal immigration.
The U.S. -Mexico border is about 3,100-km long and traverses all sorts of terrain from empty, dusty desert to the lush and rugged surroundings of the Rio Grande.
"A nation without borders is not a nation", Mr Trump said.
Fully half of the estimated 11 million undocumented immigrants living in the United States came here on legal visas and stayed after those visas expired.