U.S. Secretary of State Rex Tillerson, before the hearing of the Senate Foreign Relations Committee | Brendan Smialowski/AFP via Getty Images Tillerson signals trouble for Senate’s bipartisan Russia sanctions deal Agreement to stay tough on Russia would shut off communications with Moscow, secretary of state said.

As the Senate chugs toward a bipartisan deal to punish Russia and restrict President Donald Trump from any attempt to ease sanctions, his administration and House Republicans are signaling that the agreement has a shaky future.

Senators in both parties have urged Trump to avoid leveling any veto threat on the Russia sanctions deal that emerged late Monday, which sets up a congressional review process if the president decides to ease or remove penalties against Moscow.

But Secretary of State Rex Tillerson said Tuesday that the agreement to stay tough on Russia would shut off communications with Moscow that he'd like to keep open for now.

"What we would like is the flexibility to turn that heat up when we sense that our efforts" are failing to secure greater cooperation from Russia on anti-terrorism efforts and resolving the Syrian civil war, Tillerson told the Senate Foreign Relations Committee.

Tillerson added that the U.S. and Russia "have some channels that are open where we're starting to talk, and I think what I wouldn't want to do is close the channels off with something new."

Trump has told people since taking office that he did not intend to lift sanctions imposed in 2014 against Russia over its aggression in Ukraine, but his administration also has floated such relief as a possible incentive to get Vladimir Putin's government's cooperation on fighting terrorism. He has also been dismissive of reports of Russia's role in attempting to interfere with the 2016 election.

The House has yet to commit to taking up the Senate's bipartisan Russia deal, which would add new sanctions as well as establish a congressional review of any Trump administration changes to the current penalty structure. A House GOP aide said Tuesday that once the Senate sends a bill over, the chamber would review it.

House Foreign Affairs Chairman Ed Royce (R-Calif.) said at a committee hearing last month that he is "working to try to develop bipartisan support" for proposals to hit back at Putin's government after Russian cyberattacks aimed at disrupting the 2016 election.

Royce underscored that sanctions against Russia are already in place but added that "we are looking at ways of sending an additional message." The Senate's Russia agreement, set for a key procedural vote Wednesday, also puts the existing sanctions into law, likely making them more difficult for the White House to ease.

Sabrina Singh, spokeswoman for the Democratic opposition group American Bridge, slammed Tillerson for his Tuesday comments.

"For an administration that claims to put 'America First' they seem to go out of their way to do the exact opposite," Singh said by email. "Opposing bipartisan sanctions is sadly just the latest example of this administration prioritizing Russian interests at our country's expense."