With polls showing that Democrats could win control of the House and possibly the Senate in November congressional elections, some lawmakers say they would prefer to wait to pass Russia sanctions until Democrats have more power over what many see as unprecedented legislation.
The legislation advanced with overwhelming bipartisan support for procedural votes in the U.S. Senate on Tuesday and Wednesday, with lawmakers calling it a tribute to Graham.
Congressional aides said it could well pass the full Senate before senators head home to their districts next week.
But the legislation, renamed in Graham's honor, faces bigger challenges in the House of Representatives, which is on recess until August 31.
Already, some senior Democrats have lined up to oppose the sanctions bill.
Senator Ron Wyden and Representative Richard Neal, the top Democrats on the respective Senate and House trade committees, have blasted the bill for granting Trump new tariff powers that they say could fuel U.S. inflation.
"There is no question that the U.S. government must take stronger action against purchasers of Russian energy who are fueling the unjustifiable war against Ukraine. But the latest draft of the Sanctioning Russia Act is a prescription for bedlam and higher tariffs," they said in a joint statement.
The legislation is also driving concern among some Republicans.
Mahur Patel, a former Republican congressional aide now with Hogan Lovells Cadwalader law firm, said he doubted the legislation could pass the House this year, given the coming election season and lingering Republican skepticism about the breadth of Trump's tariffs thus far.
Patel said many Republicans in Congress remained uneasy about Trump's previous sweeping tariffs, and some were actively lobbying for specific exemptions, such as the one granted this week that offers temporary relief for phosphate fertilizer from Morocco to avert a shortfall caused by the Iran war.
Senator Rand Paul, the sole Republican to vote against the bill on Tuesday, has said that he will offer an amendment to put limits on the tariffs that Trump could impose, and others may follow suit.
Ukraine's ambassador to the U.S., Olha Stefanishyna, on Thursday said Ukraine needs quick action before the coming winter.
"You know very well that tariffs (are) a partisan issue at this point, so we hope it will not distract everyone from the real intention with the law," she told an event hosted by the Christian Science Monitor.