Expectations that President Donald Trump’s election would lead to a rapid U.S. rapprochement with Russia are fading, as the White House pushes off the Kremlin’s proposals for a meeting with President Vladimir Putin and takes an increasingly skeptical view of reaching a grand bargain with Moscow.

At a White House meeting this week, high-level national-security officials discussed possible approaches to Russia, looking to nail down elements of administration policy before Secretary of State Rex Tillerson’s first official trip to Russia in mid-April, administration officials said.

While still focusing on specific areas of possible cooperation, top administration officials now see major impediments to a broad deal on an array of policies, given Russia’s continued provocations in terms of weapons deployments, overtures to Iran, cyberintrusions and intervention in Ukraine.

The burgeoning U.S. investigations into Russia’s alleged interference in last year’s U.S. presidential election—including an inquiry into whether associates of Mr. Trump collaborated with the Russian government to help the president’s campaign—have further constrained prospects because relations with Russia are under intense scrutiny.

The Republican and Democratic U.S. senators leading an investigation into the alleged election interference plan to hold a hearing on Thursday, and both vowed at a news conference Wednesday to “go wherever the intelligence leads us.” A House probe into the same allegations has bogged down amid partisan acrimony after the panel’s chairman, Rep. Devin Nunes of California, held a meeting on White House grounds and then briefed Mr. Trump, a fellow Republican, on wiretapping without consulting committee members.

A senior administration official said Mr. Trump is “looking for and at potential areas of cooperation” with Russia but is increasingly mindful of the country’s actions that are contrary to U.S. interests. “We don’t want to be in a position where we do a lot of giving and don’t get a lot in return,” the official said.

That contrasts markedly with Mr. Trump’s sunny predictions that he could craft a new U.S. relationship with Russia and Mr. Putin. U.S.-Russia relations have significantly soured in the last four years.