WASHINGTON — The United States and Japan may fall short of signing a trade deal this week, as negotiators from both countries grapple with how to resolve President Trump’s threat to place tariffs on cars from Japan.

The two countries had been working toward signing a limited trade deal this week, as Mr. Trump and Prime Minister Shinzo Abe of Japan prepare to appear side by side at the United Nations General Assembly meeting in New York. Instead, the two sides may issue a joint statement, and keep working toward completing an agreement in the coming weeks, several people familiar with the negotiations said.

A main sticking point is Mr. Trump’s threat to put a tax on cars that are imported into the United States from Japan. The levies would be similar to those he has already placed on steel and aluminum imported from Japan, Europe and other nations. Mr. Trump has long seen his threat to tax cars, which make up more than one-third of the goods Japan shipped to the United States last year, as a source of leverage that has brought Japan to the trade negotiating table.

But that threat — and Mr. Trump’s mercurial negotiating strategy — has also become an obstacle to the deal’s resolution, according to people familiar with the discussions. Japan is seeking a firm commitment from the administration not to tax Japanese cars and is pushing to include a “sunset clause” in the pact that would cause the deal — and any benefits it has delivered to American agricultural producers — to expire if Mr. Trump follows through on his car tax threat, these people said.