For weeks, Senator Mitch McConnell, the Republican leader, has remained conspicuously on the sidelines, insisting that it was up to President Trump and Democrats to negotiate an end to the partial shutdown of the federal government.

But with the shutdown soon to enter its third week, and Mr. Trump dug in on his demand for $5 billion to build a border wall, Mr. McConnell for the first time is facing pressure from members of his own party to step in to resolve the stalemate that has left 800,000 federal workers either furloughed or working without pay.

By absenting himself, Mr. McConnell had hoped to push the blame for a prolonged shutdown onto Democrats while protecting Republicans running for re-election in 2020 — including himself. Much as Democrats did in 2018, Republicans will face a difficult map in 2020, with a handful of incumbent senators facing re-election in swing states or states won by Hillary Clinton in the 2016 presidential race.