Improving U.S.-Russian relations in 2019 will be vital to ensuring “international security,” Vladimir Putin told President Trump over the weekend in a holiday message that stressed openness and cooperation amid growing tension between the two nuclear powers.

In the letter, described by the Kremlin as a “Christmas and New Year greeting message,” Mr. Putin struck a conciliatory tone.

“Vladimir Putin stressed that Russia-US relations are the most important factor behind ensuring strategic stability and international security, and reaffirmed that Russia is open to dialogue with the United States on the most extensive agenda,” the Kremlin said in a brief description of the letter.

Russia’s state-run TASS news agency said Mr. Putin also sent a holiday letter to former President George W. Bush, but did not mention any such message to former President Obama.

Mr. Putin also sent holiday messages to the secretary general of the United Nations, Pope Francis, Queen Elizabeth II, Israeli Prime Minister Benjamin Netanyahu, Chinese President Xi Jinping, and a host of other world leaders and prominent figures.

Mr. Putin also sent a letter to Syrian dictator Bashar Assad, vowing that Moscow will continue providing military assistance to the Syrian regime in its fight against rebel forces. Russia’s influence in Syria is likely to grow in the coming months following Mr. Trump’s surprising decision earlier this month to withdraw all U.S. forces from the country.

“He stressed that Russia will continue to provide all-round assistance to the government and people of Syria in their fight against terrorism and efforts to protect state sovereignty and territorial integrity,” the Kremlin said in a brief description of the message from Mr. Putin to Assad. “The Russian leader wished the Syrian people the earliest return to peaceful and prosperous life.”

Mr. Putin’s good-natured holiday greeting to Mr. Trump comes just days after the Russian president boasted that Russia has successfully tested its “Avangard” hypersonic nuclear weapon that could easily evade all American missile defense systems.

“From next year, 2019, Russia’s armed forces will get the new intercontinental strategic system Avangard,” Mr. Putin said last week. “It’s a big moment in the life of the armed forces and in the life of the country. Russia has obtained a new type of strategic weapon.”

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