A trade deal will need to be announced this week or will likely have to wait until next year, according to Trump administration officials.
Talks are poised to spill into next week, pushing up against the goal for a deal by the end of the month.
U.S. President Donald Trump tweeted on Saturday that the United States could reach a "big Trade Agreement" with Mexico imminently.
Jesus Seade, an economic advisor to Mexico's incoming president, Andres Manuel Lopez Obrador, has been participating in portions of the NAFTA talks.
Seade predicted Saturday that the nations will also agree on a method to update NAFTA without the threat of a so-called "sunset clause", an automatic expiration after five years - as the USA has sought since October.
The deal created a trilateral trade bloc between the U.S., Mexico and Canada.
The peso initially erased its advance following the report and traded 0.6 per cent higher at 18.9005 per us dollar on Friday afternoon.
"We are well advanced (but) not there yet", said Guajardo.
The President has repeatedly said he would withdraw from the pact if it's not changed to the advantage of the U.S.
The U.S. and Mexico have been holding bilateral talks for weeks, attempting to resolve certain issues before three-country NAFTA negotiations resume. He said this week it was not a "substantive" matter and that Lopez Obrador's team had wanted to check the terms were consistent with the Mexican constitution.
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Meanwhile, Foreign Minister Chrystia Freeland of Canada said earlier in the week that she was encouraged by the progress and would rejoin the talks once bilateral discussions concluded.
Seade also said on Saturday that a "correct focus" on NAFTA's energy chapter has already been substantially agreed.
"Nothing is done until everything is truly done", he said.
Mexican Economy Minister Ildefonso Guajardo has consistently said that he expects the issue to be one of the final ones to be ironed out, because it requires Canada's participation.
"Today will be an important day", he added.
Long skeptical of foreign companies entering the Mexican oil industry, Lopez Obrador opposed Pena Nieto's energy reform.
Trump prompted the NAFTA revamp more than a year ago, complaining the pact has benefited Mexico to the detriment of United States workers and manufacturing. Business-friendly aides back greater outside investment in the industry, while his more nationalist allies want the oil to remain in Mexican hands. He made renegotiating NAFTA one of his top campaign pledges.
U.S. Trade Representative Robert Lighthizer has pushed back, and Seade has been shuttling back and forth between Washington and Mexico City to try to smooth out the issues, according to the people.
But after a year of intense negotiations aimed at salvaging the 25-year-old pact that Trump has previously referred to as a "disaster", the USA and Mexico now appear to be close to a point where Canada can rejoin the talks.