President Donald Trump’s administration tested a tough-on-Russia stance in key meetings Thursday in Europe, as the president took a new tack of his own by saying the current U.S. political climate was poisoning the prospect of better ties with Moscow.

Mr. Trump has long signaled his admiration for Russian President Vladimir Putin, raising expectations within Russia’s political class that a new U.S. administration would move quickly to roll back sanctions imposed after Moscow’s annexation of Crimea from Ukraine in 2014 and...