The White House is set to put pressure on the U.S.'s trading partners this week to sign new tariff deals, with the hope that they will be finalized before a Wednesday deadline.

The Trump administration is expected to send dozens of letters to countries that have not made a trade deal, warning them that higher tariffs could kick in Aug. 1, White House National Economic Council director Kevin Hassett and Treasury Secretary Scott Bessent said Sunday.

"Our smaller trading partners could become much bigger trading partners," Hassett said on "Face the Nation with Margaret Brennan." "And that's, I think, one of the reasons why countries are racing to set deals up with us ahead of the deadline."

As many as 15 letters will be going out beginning Monday, President Trump confirmed to reporters before heading back to the White House from his New Jersey club on Sunday. Some will go out on Tuesday and Wednesday, he said, adding that "some deals have been made."

Mr. Trump wrote later on social media that "Letters, and/or Deals, with various Countries from around the World" would start being delivered at 12:00 a.m. ET Monday.

"Any Country aligning themselves with the Anti-American policies of BRICS, will be charged an ADDITIONAL 10% Tariff," Mr. Trump wrote in another post, referring to an alliance of nations with emerging economies, which held a summit in Brazil over the weekend. "There will be no exceptions to this policy. Thank you for your attention to this matter!"

Initially comprised of Brazil, Russia, India, China and South Africa, the group now also includes Iran, Egypt, Ethiopia and the United Arab Emirates.

Mr. Trump did not specify what exactly would trigger the added BRICS tariff.  

Commerce Secretary Howard Lutnick also told reporters Sunday that higher tariffs could go into effect Aug. 1 for countries that don't make a deal by Wednesday.

Questions remain about which countries will be notified, whether anything will change in the days ahead and whether Mr. Trump will once more push off imposing the rates.

In an interview with CNN on Sunday, Bessent said about 100 letters will be sent to small countries "where we don't have very much trade" and where tariffs are already at the baseline of 10%. The treasury secretary also said that the administration is primarily focused on nations with the largest trade deficit with the U.S.

"There are 18 important trading relationships that account for 95% of our deficit, and those are the ones we're concentrating on," Bessent said.

Treasury Secretary Scott Bessent rejected the idea that Aug. 1 was a new deadline and declined to say what might happen Wednesday.

"We'll see," Bessent said on CNN's State of the Union. "I'm not going to give away the playbook."

Mr. Trump and his top trade advisers say he could extend the time for dealmaking but they insist the administration is applying maximum pressure on other nations. Hassett said that Trump would decide when it was time to give up on negotiations.

Stephen Miran, the chair of the White House Council of Economic Advisers, likewise said countries negotiating in good faith and making concessions could "sort of, get the date rolled."

The steeper tariffs that Trump announced April 2 threatened to overhaul the global economy and lead to broader trade wars. A week later, after the financial markets had panicked, his administration suspended for 90 days most of the higher taxes on imports just as they were to take effect, but he kept the 10% baseline in place while raising the tariff rate on goods imported from China. Chinese goods are now subject to 30% across-the-board tariffs.

With the freeze on sweeping U.S. tariffs set to expire Wednesday, July 9, the White House 
has touted new trade agreements with countries in recent weeks, like China, the U.K. and Vietnam. But there are still questions heading into the week with other deals yet to be announced.

Bessent said the U.S. was "close to several deals," and predicted several big announcements over the next few days. He gave no details.

"I think we're going to see a lot of deals very quickly," Bessent said.

Given the complexity of trade deals, some experts think the U.S. is likely to extend the tariff pause for some nations. 

"It can take a lot more time [than 90 days] to truly iron these things out," Clark Packard, a trade policy expert and research fellow at the Cato Institute, a nonpartisan public policy think tank, told CBS News' MoneyWatch.

Mr. Trump last week announced a deal with Vietnam. Under the terms of the agreement, Mr. Trump said the U.S. would levy 20% tariffs on goods imported to the U.S. from Vietnam, and a 40% tariff on "any transshipping." The president said Vietnam would "'OPEN THEIR MARKET TO THE UNITED STATES,' meaning that, we will be able to sell our product into Vietnam at ZERO Tariff."

That was a decline from the 46% tax on Vietnamese imports he proposed in April — one of his so-called "reciprocal tariffs" targeting dozens of countries with which the U.S. runs a trade deficit.

Asked if he expected to reach deals with the European Union or India, Trump said Friday that "letters are better for us" because there are so many countries involved.

"We have India coming up and with Vietnam, we did it, but much easier to send a letter saying, 'Listen, we know we have a certain deficit, or in some cases a surplus, but not too many. And this is what you're going to have to pay if you want to do business in the United States."

Canada, however, will not be one of the countries receiving letters, Trump's ambassador, Pete Hoekstra, said Friday after trade talks between the two countries recently resumed.

"Canada is one of our biggest trading partners," Hoekstra told CTV News in an interview in Ottawa. "We're going to have a deal that's articulated."

Canadian Prime Minister Mark Carney has said he wants a new deal in place by July 21 or Canada will increase trade countermeasures.

Hoekstra would not commit to a date for a trade agreement and said even with a deal, Canada could still face some tariffs. But "we're not going to send Canada just a letter," he said.