WASHINGTON—The White House said it would assess 10% tariffs on a further $200 billion in Chinese goods, deepening the dispute with Beijing, while sending a message to other trading partners that the U.S. won’t back away from trade fights.

The new round of tariffs—hitting products from fish to luggage—comes on top of two others and is bound to be met with threats of retaliation from Beijing, though U.S. Trade Representative Robert Lighthizer said he was open to talks with China about a resolution of the dispute.

“As...