Privately, other White House officials said that although the president would not publicly commit to signing the bill until seeing the final version, they saw no politically viable alternative if it arrived at his desk as currently written. So Ms. Sanders seized on the changes made to lay the predicate for his expected signature.

In reality, while the changes made the measure somewhat more palatable to the White House and to energy companies that objected, they mainly provided a way for the president to back down from a confrontation he was sure to lose if the sanctions bill reached the floor of the House. The Senate passed the original version of the bill, 97 to 2, and the new version, which also includes sanctions on Iran and North Korea, may come to a vote in the House as early as Tuesday.

“In the end, the administration will come to the conclusion that an overwhelming majority of Congress has, and that is that we need to sanction Russia for their meddling in the U.S. election,” Senator John Thune, Republican of South Dakota, said on “Fox News Sunday.” “That, I think, will pass probably overwhelmingly again in the Senate and with a veto-proof majority.”

Senator Benjamin L. Cardin, Democrat of Maryland and a longtime leader in pressing for more sanctions on Russia, particularly for human rights abuses, put it bluntly on the same program. “If he vetoes the bill,” Mr. Cardin said, “we will override his veto.”

Russia has bristled at American sanctions for years, particularly since the United States began imposing them under President Barack Obama in 2014 after Moscow’s annexation of Crimea and intervention in eastern Ukraine. Donald Trump Jr., the president’s eldest son, said Russian visitors with Kremlin ties raised separate human rights sanctions at a meeting during last year’s campaign, and his father said Mr. Putin raised them with him this month during a summit meeting in Germany.

The Kremlin said over the weekend that it took an “extremely negative” view of the new congressional measure but sought to dismiss the impact of its provisions. Russian news outlets noted on Sunday that the bill appeared less severe than feared.

“Vesti Nedeli,” the flagship news program of Rossiya 1, a state-owned television channel, gave only a brief summary of the new legislation, focusing instead on the Obama administration’s seizure in December of two Russian diplomatic compounds in Maryland and New York.