McConnell, up for re-election in 2020 in a state where Trump is more popular than the majority leader, has been largely sidelined during the shutdown, despite a reputation as a canny negotiator.
After Trump refused to accept a spending bill that the Senate passed in December by 96-0 on a voice vote, McConnell insisted that the Senate will only vote for a spending bill that Trump will sign.
But as the government shutdown threatens to set a new record, a number of Senate Republicans up for re-election in states that Trump lost in 2016 have urged Republicans to quickly get the government up and running.
Sen. Cory Gardner, R-Colo., last week called for re-opening the government even without a deal on Trump's wall. Sen. Susan Collins, R-Maine, who could also face a challenging re-election said she'd like to see Trump support at least part of the House Democrats' plan to reopen the government.
Republican senators involved in current discussions with McConnell -- led by Graham -- include Collins, Tillis, Lisa Murkowski of Alaska, Lamar Alexander of Tennessee and Rob Portman of Ohio. At Graham's office meeting Wednesday night, Trump's son-in-law and senior adviser, Jared Kushner, was also in attendance, suggesting he might also be interested in the group's gambit.
Senate Democrats this week have sought to increase the pressure on Senate Republicans and McConnell, a master of Senate procedure and nuance, by blocking votes on any legislation unrelated to the government shutdown. McConnell tussled on the Senate floor with Democrats before meeting with the Senate Republicans.
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"The Senate itself is being shut down because of colleagues' refusal to do business," McConnell complained. "There is no precedence for that, there's no reason, we're all here."
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