"The stakes are so high here," said Brady, who has been involved in a dozen free trade negotiations "in one way or another." He led a bipartisan delegation to Mexico to meet with NAFTA negotiators during the past two days.
In an interview on CNBC's "Squawk Box," the Texas Republican said negotiators are making "good progress" and they're "close on some major economic wins." He added that he didn't see anything while visiting the U.S. negotiating team to make him believe the U.S. would abandon NAFTA.
Trump's announcement on Thursday of plans to impose broad import tariffs on steel and aluminum has added a layer of complication to the NAFTA talks. The stock market was slammed on Thursday and Friday on fears of a trade war. But Monday and early Tuesday, some of those concerns abated.
On Sunday, Brady said all fairly traded steel and aluminum — especially from Canada and Mexico — should be excluded from the tariffs.
While there's been no confirmation from the White House on whether such exemptions are in the works, the president this week signaled that could happen if a new, "fair" NAFTA can be reached.
Trump also said Monday he would stick to his tariff plan despite growing pressure from Republican leaders on Capitol Hill, including House Speaker Paul Ryan. Ryan's home state of Wisconsin would be hit by proposed European counter-measures on Milwaukee-based Harley-Davidson.
Meanwhile, there's speculation that Trump may consider additional tariffs on cars imports. European automakers are urging the president to abandon thoughts of imposing a retaliatory tax on vehicles coming into the U.S. from the European Union.
"This may well be a negotiating approach" on the part of the president," Brady told CNBC on Tuesday. But he added that Trump is right to want to level the playing field in the steel and aluminum industries, which are rife with unfair trading terms for U.S. producers.
The House Ways and Means chairman, first elected to Congress in 1996, has had a pretty good track record predicting tricky outcomes lately, considering he kept saying against what looked like long odds at many stages that GOP tax reform would pass.
As the top tax writer in the House, he took the lead in crafting and pushing through the new Republican tax law that overhauls the brackets for individuals and cuts the rate for U.S. corporations.
— Reuters contributed to this report.