The president-elect said the Japanese company would be hit with a "big border tax" if the plan went ahead.
American car companies have faced harsh criticism from Mr Trump for building cars more cheaply outside the US.
Toyota's President Akio Toyoda said the company had no immediate plans to curb production in Mexico.
"We will consider our option as we see what policies the incoming president adopts," said Mr Toyoda, speaking in Japan on Thursday.
The company's US arm issued a statement saying production and employment levels at Toyota in the US would not decrease as a result of the new plant in Mexico. The company has 10 manufacturing plants in the US.
"Toyota looks forward to collaborating with the Trump Administration to serve in the best interests of consumers and the automotive industry," the statement said.
The carmaker's shares fell more than 3% in early trade in Tokyo on Friday but have since recovered some ground.
Japan's trade minister Hiroshige Seko,  speaking at a regular news conference on Friday, said the new US administration needed to understand that his country's auto industry "has greatly contributed to the US economy".
Mr Trump has already targeted American carmakers General Motors and Ford for manufacturing south of the border.
Ford later cancelled its plans for a $1.6bn (Â£1.3bn) plant in Mexico and said it would expand operations in the US instead, but said this was due to market considerations.
Currently there are no tariffs on goods passing between the US and Mexico, as the countries - and Canada - agreed to trade freely without tariffs under the Nafta trade pact more than 20 years ago.
But Mr Trump has said he wants to renegotiate Nafta and has threatened to withdraw from it if he cannot get a better deal for the US.
Toyota announced in April 2015 it would build a $1bn Corolla factory in central Mexico. Construction began in November 2016.
It has existing manufacturing facilities in Baja, Mexico which were established in 2002.
Donald Trump has tweeted that Toyota will face hefty tariffs on cars built in Mexico for the US market if it builds Corollas south of the border.