“I don’t think it’s a question at all that our relationship with Turkey would be weakened at all by this,” Ms. Trudeau added.

She said she had no update on Turkey’s request for the United States to extradite Mr. Gulen, a “legal, technical process” about which American officials have been in direct contact with the Turkish authorities. The administration, Ms. Trudeau said, feels anti-American rhetoric is “unhelpful” to the United States’ relationship with Turkey. “We believe our relations and our partnership and our friendship with Turkey is strong,” she said.

As for Russia and Turkey, Syria remains a major potential fault line, despite the pledges to work together. Mr. Putin noted that the views of the two sides “do not always coincide” when it comes to Turkey’s southern neighbor. Mr. Erdogan is a bitter enemy of Syria’s president, Bashar al-Assad, and has insisted that he step down before peace negotiations can begin. Russia, though, is a longtime ally of Mr. Assad’s, and it intervened with Iran in the Syrian conflict to bolster his fortunes.

Yet the Kremlin also signaled on Tuesday that it was in Syria to stay. Mr. Putin called on Russia’s Parliament to approve an extended deployment of the Russian Air Force at Khmeimim Air Base outside Latakia, Syria, where its planes have flown sorties for almost a year to bolster Mr. Assad. Parliamentary approval is virtually guaranteed.

“This is a demonstration that Russia has come to Syria for a very long time,” said Aleksandr M. Golts, a Russian military analyst. “This is a demonstration that it will support Assad and that it is ready to tie itself to a regime that is involved in a bloody civil war.”

Russia would like Turkey to seal its borders and stem the flow of fighters and weapons to the insurgents, and to reverse its demand that Mr. Assad must go. Ankara wants Moscow to stop bombing its insurgent allies; to lessen support for the Kurds; and to halt the bombing of civilian populations, which drives refugees into Turkey.

As a possible sign of good will, a major Kurdish representative office closed in Moscow on Sunday, although the local representative said it was because of rent costs rather than politics.