The government of Denmark has not historically been a big spender on Washington lobbying. But days before President Trump took office for a second time, Denmark’s embassy started shopping for a lobbyist with ties to the new president, who has loudly proclaimed his intention to try to take over the autonomous Danish territory of Greenland.

The Danes are not alone.

A number of countries that would be affected by Mr. Trump’s threatened acquisitions, tariffs, aid reductions or deportations have been urgently seeking help on K Street to navigate his administration.

Panama, which is pushing back against Mr. Trump’s threat to reclaim the Panama Canal, signed a contract three days before his inauguration that could pay nearly $2.5 million over the next year to a team of lobbyists including the prominent Trump ally David Urban of BGR Group, according to Justice Department lobbying filings.

BGR Group, a powerhouse firm founded by establishment Republicans in the 1990s, also signed a $600,000 one-year contract in late November with the Embassy of Somalia, which is facing the prospect that Mr. Trump’ administration might scale back military cooperation.

And South Korea, which could be hurt by tariffs, expanded its lobbying corps after the election to bring aboard the former Trump campaign adviser Bryan Lanza and his team at Mercury Public Affairs, according to lobbying filings and a person familiar with the engagement.

The surge in interest from foreign governments in lobbyists with connections to the new administration underscores the uncertainty with which even longtime U.S. allies regard Mr. Trump. The administration’s brinkmanship with Colombia over tariffs and the deportation of unauthorized immigrants over the weekend showed how quickly any tensions could escalate.