WASHINGTON — President Trump’s efforts to hide his conversations with President Vladimir V. Putin of Russia and new details about the F.B.I. inquiry into his ties to Moscow have intensified debate over his relationship with Russia, adding fuel to Democrats’ budding investigations of his presidency and potentially setting up a clash between the White House and Congress.

Representative Adam B. Schiff of California, who now leads the Intelligence Committee as part of the new Democratic House majority, implored his Republican colleagues Sunday to support his effort to obtain notes or testimony from the interpreter in one of the private meetings between Mr. Trump and Mr. Putin.

“Will they join us now?” Mr. Schiff wrote on Twitter. “Shouldn’t we find out whether our president is really putting ‘America first?’”

The administration appears unlikely to acquiesce to such a demand without a fight.

Mr. Trump has repeatedly withheld details of his conversations with Mr. Putin, according to current and former American officials, a practice that has left officials blind to the dynamic between the two leaders and intensified questions within the administration over the president’s actions.