The steep US tariffs on steel and aluminum imposed last year on national security grounds have become a major stumbling block to ratifying a new North American trade pact the three countries negotiated last year.
The comments came as Canada's Foreign Minister Chrystia Freeland was in Washington for meetings with US Trade Representative Robert Lighthizer and congressional leaders.
Following her meeting with Lighthizer, she repeated the demand the tariffs -- 25 percent on steel and 10 percent on aluminum -- be withdrawn before the implementing the revised deal known as the US-Mexico-Canada Agreement.
"It was a good meeting," she told reporters, confirming that the officials discussed the steel and aluminum levies, as well as other elements of the USMCA.
In his testimony before a Senate committee, Mnuchin said, "I think we are close to an understanding with Mexico and Canada."
US lawmakers have likewise resisted bringing the new agreement to a vote while the tariffs remain in place for the US's northern and southern neighbors, traditionally major US suppliers of steel and aluminum.
But US officials have reportedly sought to persuade Mexico City and Ottawa to accept export quotas in return for lifting the metal tariffs for their countries.
Mnuchin said Wednesday this was likely to be worked out soon. However, he did not provide any details.
"I've spoken to the finance ministers," he said. "I can assure you it is a priority for us."
Lighthizer also was due to meet with congressional leaders later Wednesday, including House Speaker Nancy Pelosi.
China talks in 'near future'
Mnuchin also repeated his expectation that US negotiators would return to Beijing to pursue stalled talks aimed at resolving the US-China trade war.
With Lighthizer, Mnuchin has led the US delegation in 11 rounds of shuttle diplomacy aimed at resolving the trade dispute between the world's two biggest economies.
But hostilities resumed last week when the United States accused China of backsliding on major commitments already made in the talks and President Donald Trump more than doubled punitive tariffs on $200 billion worth of Chinese merchandise imports.
China responded in kind on Monday, raising duties on $60 billion in US exports.
"I think we had a constructive meeting with the vice premier," Mnuchin told lawmakers Wednesday, referring to last week's brief meetings with Chinese trade envoy Liu He.
"There's still a lot of work to do," he added. "As I've said, my expectation is we will most likely go to Beijing at some point in the near future to continue those discussions and I think it's President Trump's expectation to meet with President Xi at the G20 at the end of June."