The Republican-led Congress passed a package of withering sanctions targeting Russia a year ago, in part to punish its 2016 efforts. But a debate over how to amp up pressure has blossomed anew in recent weeks, as lawmakers have grappled with new disclosures about continued Russian activity and Mr. Trump’s widely panned meeting last month with Mr. Putin in Helsinki, Finland.

Mr. Trump spent the days leading up to the meeting bashing the Atlantic alliance, which is committed to counteracting Russia’s influence. Once there, he appeared to accept Mr. Putin’s denials that Russia did not interfere in 2016 and cast doubt on American intelligence agencies’ findings on the matter.

Many Republicans, long uncomfortable with the president’s forgiving stance toward Russia, condemned Mr. Trump’s performance but have been unwilling or unable to take direct action. The possibility of additional sanctions appears to be an exception.

Senator Mitch McConnell of Kentucky, the typically reserved majority leader, called out Mr. Rubio’s bill by name last month and asked the Senate Banking and Foreign Relations Committees to hold hearings on Russia and the imposing of sanctions already on the books. Those hearings are underway, though Democrats fret that Mr. McConnell may not have any intention of actually pushing through legislation.

Still, the senators’ argument for tough immediate action received an added boost this week, when Facebook disclosed that it had detected and tried to disrupt a sophisticated influence campaign on its networks sowing discord ahead of November’s elections. Though the company said it was unable to definitively attribute the activity to Russia, the tactics bore a striking resemblance to those used around the 2016 election. And some of the accounts interacted with accounts known to be operated by the Internet Research Agency, the Kremlin-linked group that was at the center of an indictment related to earlier interference.

Senator Claire McCaskill, Democrat of Missouri, also confirmed a Daily Beast report that her office was unsuccessfully targeted by Russian hackers last year. American intelligence officials have indicated that she is not alone, but no other office or candidate has been publicly identified.

The sanctions package is intended to mount pressure on Russia to stop such activity. It includes a long list of provisions meant to embarrass Mr. Putin, cripple his allies financially and create new criminal vulnerabilities for Russia and other nations participating in cyberattacks. It would mandate that the United States assemble a report on Mr. Putin’s finances and assets — a move long advocated by Russia hawks who believe it would expose the extent of any financial crimes committed by Mr. Putin. And it penalizes projects involving Russian sovereign debt and state-owned energy companies — a move that some Republicans, including in the Trump administration, fear could wreak havoc across global markets.