Washington (CNN) President Donald Trump's hasty announcement last week that he'll slap new tariffs on steel and aluminum imports has prompted a public free-for-all to finalize details of the plan, with efforts underway to assuage some concerns from within Trump's own party.

Trump insisted on Monday that he wouldn't back off his promise to apply a 25% tariff on steel and a 10% tariff on aluminum. But there are indications the plan isn't yet finalized and that whatever is ultimately announced may be narrower in scope than initially previewed.

Speaking with Prime Minister Justin Trudeau on Monday evening, Trump signaled he was flexible in how the proposed tariffs would apply to Canada, a person close to the trade talks said. During a conversation with British Prime Minister Theresa May on Sunday, Trump indicated he had not made a final decision on what to do with steel and aluminum tariffs, according to person familiar with the call.

In his talks with foreign leaders over the past several days, Trump stopped short of making any firm commitments, the sources said. He listened carefully to his counterparts' arguments against the plan, which has also been opposed by leading Washington Republicans.

"As the President just announced, Canada is a very significant partner that buys steel and sells steel. To extent that we're successful in renegotiating NAFTA those tariffs won't apply to Mexico and Canada. But we look forward to the president releasing the specific details and working with other people," Treasury Secretary Steve Mnuchin told a congressional panel on Tuesday.

But the conversations are a sign that Trump's announcement, made only when pressed during a meeting with steel industry executives last week, wasn't fully solidified when it was made. The loose details have allowed opponents of the tariffs -- including Trump's own aides -- an opening to alter Trump's thinking.

Rivals of the tariffs inside the White House are compiling daily clips of Republican criticism to present to the President, hoping to balance out the loud voices arguing in favor of the plan. Republican governors, such as Scott Walker of Wisconsin, have advised Trump to rethink his decision. Economists that Trump respects, such as former campaign adviser Stephen Moore and CNBC contributor Larry Kudlow, have said the tariffs could damage the economy. House Speaker Paul Ryan has engaged in a public dispute with the President on the matter.

But supporters of the plan, including trade adviser Peter Navarro, have argued equally as loudly that the tariffs would please Trump's conservative base and Trump has signaled he's unwilling to budge, at least in principle.

"No, we're not backing down," Trump told reporters in the Oval Office on Monday.

The debate comes as a special election approaches in Pennsylvania steel country. Defending the Republican-held seat is a chief priority for Trump, who has told advisers his steel announcement would satisfy voters there. He's due to campaign for the GOP candidate, Rick Saccone, on Saturday, and is widely expected to tout his decision on steel. But the uncertainty surrounding the actual order imposing the new tariffs has led to doubts the plan will be formally inked by then.

Some White House aides, including Gary Cohn, are working to organize a meeting between Trump and representatives from the auto and bottling industries to fully explain the effect of steel and aluminum tariffs, according to a person familiar with the plan. The meeting is tentatively set for Thursday, but a White House official said nothing is firmly set on the schedule.

But Republican Sen. Jeff Flake told CNN on Tuesday Cohn's voice doesn't seem to be pulling much weight in the White House tariff debate.

"Gary is right on this, but it doesn't seem as if his opinions are holding sway. I don't know. This is one that the President just keeps coming back to," Flake said.

The Arizona Republican said he is doubtful that any legislation to limit Trump's authority on trade would move "under the circumstances."

"I seriously question whether you'll have enough Republicans senators want to cross the President this way," Flake said.

The meeting, in the minds of its organizers, would help fully illustrate the economic pitfalls of tariffs on American industries. Supporters of the tariffs -- including Commerce Secretary Wilbur Ross -- have said the effects on consumers would be negligible. Ross colorfully illustrated his point using a can of chicken noodle soup on television last week.

Ross and Navarro have represented a hard-line view on tariffs during Oval Office conversations over the past year. They've been pitted against Cohn, who has ardently argued that imposing tariffs would prove catastrophic.

Last week, Cohn told some associates that he was prepared to leave the White House if the new tariffs materialize. But he has held out hope the plan can be suitably mitigated and is actively working to convince the President that sweeping tariffs on steel would slow a booming economy.

He's been backed by Ryan, who explained his approach on Tuesday.

"What we're encouraging the administration to do is to focus on what is clearly a legitimate problem and to be more surgical in its approach so we can go after the true abusers without creating any kind of unintended consequences or collateral damage," Ryan said during a news conference.

Officials say Trump has listened carefully to the criticism of his announcement and concede it's likely that the final plan won't apply the same tariffs to every country.

"I think he's looking at the arguments to make those decisions," one official said.

Another administration official said it is likely there will be some country exclusions -- and the burden of making that case will largely rest on Defense Secretary James Mattis and national security adviser H.R. McMaster's shoulders.

Those advisers have repeatedly argued for certain country exclusions on national security grounds. The United Kingdom, for example, is the only source of a specialty steel the US uses in its submarine production.

Trump himself said on Monday that Canada and Mexico could be excluded from the tariffs if they agree to a "new & fair" North American Free Trade Agreement, currently the subject of negotiation in Mexico City.

We have large trade deficits with Mexico and Canada. NAFTA, which is under renegotiation right now, has been a bad deal for U.S.A. Massive relocation of companies & jobs. Tariffs on Steel and Aluminum will only come off if new & fair NAFTA agreement is signed. Also, Canada must.. — Donald J. Trump (@realDonaldTrump) March 5, 2018

But country exclusions could lead to an increase in the tariff rate for the rest of the world in order to achieve the same effect on the American domestic steel aluminum industries, a point Navarro underscored on Sunday morning television shows.

Lawyers in the Justice Department's Office of Legal Counsel and the White House counsel's office had told White House officials last week -- before Trump stunned them with his announcement -- that it would take "weeks" before the tariff proclamation authority would be ready.

Now, they are working on a sped-up timeline. Administration lawyers that had been helping the White House with other matters have been pulled off those projects and are now solely focused on the tariff proclamation so that it can be unveiled late this week or early next, people familiar with the matter said.