(CNN) The Trump administration spent more than a year working through high-stakes, strained negotiations for a new North American Free Trade Agreement. But that was just the beginning.

The most arduous part still remains: selling the deal to a divided Congress.

President Donald Trump's top trade officials regularly argued the US-Mexico-Canada agreement, with its new labor and auto manufacturing requirements, would be an easy sell for newly empowered House Democrats, but the task of persuading members to support it is proving more difficult than they initially suggested. Tensions from the President's unresolved tariffs on steel and aluminum from Canada and Mexico haven't made it any easier for foreign allies to rally behind the new deal, either.

There is a sense of urgency to ratify the deal as Canadian elections approach in October and the 2020 US presidential campaign gets underway. But even as the White House makes a concerted push to bring Democrats on board, lawmakers say they aren't in any rush to move things along.

"We're not going to be held to an artificial timetable," Rep. Earl Blumenauer, the chairman of the Ways and Means subcommittee on trade, told CNN on Monday. "It will move when it's ready to move. And the Democratic leadership has shown in the past that they'll pull the plug, if necessary, if things aren't resolved in a reasonable fashion."

The Oregon Democrat described the trade deal as "a work in progress" and estimated it would be several months before action is taken on it.

Trump threatened in December to withdraw from NAFTA in order to force Congress to approve his deal, but he hasn't yet followed through on it.

Such a decision would face legal challenges and would alienate lawmakers on both sides of the aisle. Instead, the administration has so far opted for a different, more diplomatic tack. US Trade Representative Robert Lighthizer has made himself available on Capitol Hill, meeting with Democrats frequently to hear their concerns.

But his outreach hasn't moved votes yet. Trump reiterated his threat in an interview with Fox Business's Maria Bartiromo last week, saying his fallback would be to return to "maybe pre-NAFTA. NAFTA's one of the worst trade deals ever made."

The White House had hoped trade-skeptical Democrats and organized labor, which has long opposed the NAFTA agreement signed by President Bill Clinton in 1994, would like Trump's deal, as it includes new minimum wage and auto content rules, as well as calling for Mexico to pass significant labor reforms.

But no such bipartisan coalition has emerged. In the past month, Trump's deal has failed to win over influential labor groups including the AFL-CIO, the United Auto Workers, and most recently, the United Steelworkers.

Critics of the agreement have called for beefing up its labor and environmental protections, as well as ensuring those aspects are actually enforceable in practice. Some Democrats have asked the administration to make changes to the deal, like taking steps to lower prescription drug prices and expand the scope of the new minimum wage requirements.

Celeste Drake, the AFL-CIO's top trade and globalization policy guru, will testify before Blumenauer's subcommittee during a hearing on Tuesday morning, alongside representatives from the United Auto Workers, the United Steelworkers, and other key organized labor groups.

Democrats are waiting to see whether the Mexican government adopts the labor legislation it is obligated to pass under the USMCA. Those reforms, which would enshrine new protections for union workers, such as establishing independent courts for labor disputes, have faced opposition from some members of Mexico's Congress and the Mexican business community.

"I don't think anybody's going to run to embrace" the deal if the Mexican government does not implement the labor reforms it agreed to, Blumenauer told CNN.

Members are also reserving judgment until the International Trade Commission completes its report on the deal, which is expected in the middle of April. "Part of it is going to require people to get a better handle on what's there, what they hear back home, and what the choices are," he said.

In the meantime, Canadian and Mexican officials are placing pressure on Trump to remove his steel and aluminum tariffs.

"To impose tariffs on Canada on national security grounds is, frankly, absurd," Canadian Foreign Minister Chrystia Freeland said after a meeting with Lighthizer on Monday afternoon.

"The existence of these tariffs, for many Canadians raises serious questions about NAFTA ratification," Freeland said.