Nearly 300 U.S. Marines landed in Norway on Monday with a clear assignment from the outgoing Obama administration: Train with allies and help reinforce the North Atlantic Treaty Organization’s northernmost border with Russia.

But the latest comments from President-elect Donald Trump—who referred to NATO as “obsolete” in a newspaper interview—have left the U.S. military in an awkward spot as its new commander-in-chief signals an interest in improved ties with Moscow.

Mr. Trump’s comments—he also called for making “some good deals with Russia” and said NATO’s focus should be terrorists, not Moscow—sparked fresh unease in Europe, prompting leaders there to call for unity in the face of trans-Atlantic tensions.

Norway’s foreign minister, Børge Brende, declined to comment on Mr. Trump’s remarks but said he expected the U.S. to stick to promises made to allies.

U.S. officers said the Norway deployment is about strengthening partnerships. “This is all about assuring our partners about our willingness to support and defend them and NATO,” Marine Maj. Gen. Niel Nelson said in an interview this month.

The Marines are part of a broader effort to bolster NATO defenses in the wake of the Russian takeover of Crimea from Ukraine in 2014. A U.S. Army heavy brigade arrived in Poland last week. Its tanks are to deploy across Eastern Europe in February.

During a visit to Kiev on Monday, U.S. Vice President Joe Biden said sanctions should remain in place against Moscow until Russia has fully implemented a peace deal for Ukraine, where Russian-backed rebels hold parts of the country’s east.

For more than two years, Moscow and the Western allies have been engaged in a battle of words, while reinforcing their borders with extra troops. Each side has accused the other of escalation.

NATO created a rapid-reaction force and the U.S. put small contingents of soldiers in Poland and the Baltic countries. Russia pledged to deploy more troops on its borders and boosted naval and missile forces in Kaliningrad, between Poland and Lithuania.

The Russian moves were followed by a NATO decision to create a 4,000-person deterrent force, and by the U.S. to move additional tanks, artillery and helicopters to Europe.

At issue in Norway is whether the country is breaching the so-called base agreement, a decades-old pledge to Moscow that Oslo wouldn’t open its bases to foreign troops unless it was attacked or threatened.

In a 1997 agreement with Moscow, which paved the way for NATO expansion, the alliance also pledged not to permanently station substantial numbers of combat troops on Russia’s borders.

When the Pentagon released details of the Norwegian training last fall, the Russian embassy in Oslo warned it “certainly won’t improve security in Northern Europe.”

Norwegian Defense Minister Ine Eriksen Søreide said the American soldiers’ arrival doesn’t mark a shift in Norwegian security policy. Training alongside NATO allies has been a central part of the Norwegian military’s strategy for decades, she said.

The Marine deployment “does not give grounds for any reaction from Russia,” said Ms. Søreide.

Inside Norway, however, some have protested the NATO deployments, saying they risk plunging the region into a dangerous cycle of military escalation.

Hedda Langemyr, head of the Norwegian Peace Council, said hosting U.S. troops in the Nordic country could have a polarizing effect on Russia. “This increases the chances of confrontation in the North and is not a particularly wise strategy for Norway,” she said.

The Marines will be focused on conducting joint exercises with Norway, Romania and Lithuania, according to Gen. Nelson, the commander of U.S. Marine Corps Forces in Europe and Africa.

The U.S. and Norway have agreed to two six-month-long rotations of Marines as a pilot project, and will decide later in the year whether to expand the deployment or discontinue it.

Gen. Nelson said the Marines will conduct exercises at a training ground just 120 miles from Norway’s border with Russia, gaining expertise on how to survive and fight on the tundra.

“It is quite a different environment to wake up in 6 feet of snow than it is to wake up in the desert,” Gen. Nelson said. “Our folks are leaning a whole variety of human lessons.”