WASHINGTON — Mexico hit back at the United States on Tuesday, imposing tariffs on around $3 billion worth of American pork, steel, cheese and other goods in response to the Trump administration’s steel and aluminum levies, further straining relations between the two countries as they struggle to rewrite the North American Free Trade Agreement.

The tariffs, which were announced last week, came into effect as the Trump administration threw yet another complication into the fractious Nafta talks. Officials are now saying they want to splinter discussions with Canada and Mexico and work on separate agreements rather than continue three-country discussions to rewrite the 1994 trade deal.

Larry Kudlow, President Trump’s chief economic adviser, said on Tuesday that Mr. Trump’s “preference now, and he asked me to convey this, is to actually negotiate with Mexico and Canada separately.”

Mr. Kudlow, speaking on “Fox and Friends,” said pursuing separate deals might allow an agreement to be reached “more rapidly,” adding: “I think that’s the key point. You know, Nafta has kind of dragged on.”