McConnell also urged Americans to take comfort in the record number of circuit court judges the Senate has managed to confirm under the Trump administration, despite Democrats' slow-walking. It's something Corker acknowledged to reporters when he noted that "doing anything up here is like pushing a major boulder uphill". They plan to push it through their think-tank, grassroots political network, and Hispanic libertarian group.
Those are just some of the reactions from Senate Republicans since last week in light of the administration's decision to impose tariffs on imports of steel and aluminum from the European Union, Canada, and Mexico.
According to McConnell, the only possibility would be if there were legislation President Donald Trump were to sign. He was heading to the White House later Tuesday for talks. It strictly prohibits the Pentagon from working with any entity that uses telecommunications equipment or services produced by Huawei Technologies Company or ZTE Corporation, which is facing trade law violations over selling sensitive technologies to USA adversaries.
Rep. Tom Cole, R-Okla., said it's too early to say whether the House would take up Corker's proposal.
Republican leaders have balked at the idea of using legislation to stop Trump with House Speaker Paul Ryan - who opposed aluminum and steel tariffs last week - telling reporters Wednesday morning that there was little appetite to pass legislation in the House similar to Corker's.
The bipartisan legislation that Corker filed would require the president to submit to Congress any tariffs put in place under a process that invokes national security.
He also said that the US steel and aluminum industries have "been ravaged" by foreign trade policies, adding that "this is only the first step".
The defense bill will be open for amendments, McConnell said, and "we'll see what happens as it moves across the floor".
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Only one Republican incumbent, Senator Dean Heller of Nevada, faces a similar dilemma. Now some of the biggest Republican donors have launched a war against Trump's tariffs, and once-loyal senators are trying to curb Trump's powers as the country's 45th president.
Corker isn't alone on his side of the aisle. Sen. Anyone who doesn't back their free trade principles is "going to have a harder time getting support from our network", an official with the Koch network told Quartz.
The Senate is split between 31 members of the Democratic conference and 31 members in the Republican conference.
"But Corker declined, saying he told the president: "'I'm a United States senator, and, you know, I have responsibilities, and I'm going to continue to carry them out'".
But the Democratic Party's leaders expressed support for working through August.
Democratic Leader Andrea Stewart-Cousins said she and the GOP leadership agreed that Tuesday would be the day when Democrats could present their long-awaited amendment to enact the women's reproductive health act.
Now, Senate Majority Leader Mitch McConnell wants to bring that to an end - and the timing couldn't be more suspect. Heidi Heitkamp, a North Dakota Democrat in a tough re-election fight in 2018, said she was "very interested" in the proposal.
Joni Ernst said that senators "should stay in Washington until we get our job done".