The U.S. is aiming to reach a deal in principle with Canada and Mexico on the North American Free Trade Agreement in the next three weeks, according to congressional aides and industry executives who have been briefed by the Trump administration.

In the last week, U.S. Trade Representative Robert Lighthizer told lawmakers in one-on-one phone calls to allow two weeks for high-level talks to conclude, and an additional two weeks for the fine print to be hammered out, according to aides. A senior administration official confirmed that this was the timing the U.S. expected.

The Office of the U.S. Trade Representative declined to comment.

Lighthizer's outreach has spanned both sides of the aisle; within the Ways and Means committee alone, it has included free-trade supporters like Rep. Kevin Brady, R-Texas, and skeptics like Reps. Bill Pascrell, D-N.J., and Richard Neal, D-Mass.

"I think it's encouraging that all three countries are really focused toward a late April/early May agreement," House Ways and Means Chairman Brady told reporters Monday.

The range of lawmakers that Lighthizer has contacted illustrates the need for the administration to get support for its deal from both sides of the aisle, since its trade agenda doesn't clearly reflect one party's ideology on the issue.

Illinois Republican Darin LaHood challenged Lighthizer in a March congressional hearing to name an industry group that supported the White House's more hawkish policies.

"How about the AFL-CIO, do they count?" Lighthizer replied, citing the country's largest labor federation, representing 12 million unionized workers.



Rep. LaHood followed up to ask whether any business-friendly groups – like the U.S. Chamber of Commerce or the Business Roundtable – supported proposals like those mandating a high percentage of U.S. parts in automobiles produced in Nafta countries.

"There are business groups all over the place, I have no idea where they are on rules of origin," Lighthizer said. He then suggested it was the business-friendly (and historically GOP-supported) policies pushed by those lobbies that created the trade deficits he was tasked with narrowing.

Lighthizer, Canadian Foreign Minister Chrystia Freeland and Mexican Economy Minister Ildefonso Guajardo are set to meet in Washington on April 19 and 20, according to three government officials, the second set of in-person meetings the top trade negotiators have held in the U.S. this month. But while these officials say talks continue, they caution not to expect any announcements this week.

Negotiators had been working toward a deal by May 1. President Donald Trump had exempted Mexico and Canada from steel and aluminum tariffs until that date, and had suggested that a permanent exemption would only be granted if the two countries were seen bending to U.S. wishes during Nafta talks.

"We're going to hold off on the tariffs on those two countries to see whether or not we're able to make the deal on Nafta," Trump said on March 8 while unveiling the 25 percent levy on imported steel and 10 percent on imported aluminum.

The announcement of a deal mid-May would come nearly a year after Lighthizer first notified Congress of the Trump administration's intent to reopen Nafta talks. Since then, the U.S., Canada and Mexico have held seven rounds of negotiations, tackling first the low-hanging fruit like e-commerce, which had been excluded from the original pact, and leaving the more contentious issues until the end.

All the while, Trump has threatened to withdraw – and may continue doing so, a senior administration official said, especially if members of Congress appear unwilling to ratify the deal the administration presents to them.

--CNBC's Stephanie Dhue contributed to this report.