Even as President Donald Trump and top Republicans in Congress call on Democratic leaders in the U.S. House to allow a vote on a new trade deal with Mexico and Canada, the President's top trade negotiator told Senators on Tuesday that there's still no set date for when the agreement would be submitted to the Congress

"I believe we're on track, I believe we are making progress," said United States Trade Representative Robert Lighthizer.

Asked by a GOP Senator about discussions with Speaker Nancy Pelosi, Lighthizer gave no public hint about any problems.

"My hope is that over the course of the next several weeks, that we can make substantial progress," Lighthizer added, as he said talks with Pelosi had been 'constructive.'

Democrats have been pressing the Trump Administration over the enforcement of new labor reforms in Mexico, worried that the government won't adequately enforce the changes.

Asked by Sen. John Cornyn (R-TX) when to expect a vote in Congress, Lighthizer gave no concrete date - as the trade agreement has not yet been formally submitted to the Congress.

At a hearing of the Senate Finance Committee, Lighthizer faced some verbal slings and arrows from both parties about the President's trade policies.

"I do not agree that tariffs should be the tool we use in every instance to achieve our trade policy goals," said Sen. Charles Grassley (R-IA).

"China's market is now more closed off to American goods and American agriculture than before the trade war began," said Sen. Ron Wyden (D-OR), as he complained about the impact of the President's tweets on trade policy.

For the most part, Lighthizer did not engage in pitched battles with Democrats over trade matters, repeatedly stressing common ground over trade disputes with China and final talks over the USMCA trade deal.

As for China, Lighthizer made clear that President Trump isn't bluffing when it comes to additional tariffs on Chinese goods, acknowledging to Senators that the next round could have a bigger impact, to include items like laptop computers and cell phones.

Lighthizer could have a somewhat more partisan reception on Wednesday, when he testifies on the same issues before the House Ways and Means Committee.