Mexico's Economy Minister Ildefonso Guajardo said he expected to learn by the end of Friday whether a new deal was possible. He and his counterparts have been meeting in Washington since Monday to try to bridge major gaps.
"I think we will be finding out through the day and tomorrow ... if we really have what it takes to be able to land these things in the short run," Guajardo told Reuters.
A source close to the talks said it was possible that Guajardo, U.S. Trade Representative Robert Lighthizer and Canadian Foreign Minister Chrystia Freeland could extend their meetings into the weekend.
A USTR spokeswoman declined comment while a Freeland spokesman did not respond to a request for comment.
Guajardo told Reuters that "we have suitcases for two weeks if necessary."
This week's talks hit an obstacle as the United States and Mexico sought to settle differences over the key issue of automobiles.
"Mexico obviously is here in order to negotiate the best agreement for Mexican workers and consumers. It will take as long as it will take," Mexican deputy economy minister Juan Carlos Baker told reporters late in the day.
But Ryan expressed skepticism that a deal could be reached in time and noted that several major issues remained unresolved, such as U.S. demands for more access to Canada's dairy market and to make an investment dispute arbitration system optional.
"There are a handful of unresolved issues and I'm just not — I don't want to make news, but we'll see if they can get this done by May 17 and get us the paper to Congress, which then we could have this vote in December," Ryan said. "If they can't, then we won't."
Trump regularly threatens to walk away from NAFTA, underscoring uncertainty over the pact. Business executives complain that the lack of clarity is hitting investment.