Democrats are trying to woo Republicans away from Trump's wall fight -- but the GOP is losing them
Donald Trump and House GOP members in the White House Rose Garden on Thursday (Screen capture)

As Democrats work to woo Republicans away from Donald Trump's wall fight, the GOP keeps losing them as the government shutdown continues into its third week.

Politico reported Monday that the White House is also working to win Democratic votes away from the opposition party in a "last-ditch effort" to lure in moderates with a border security package.

The newspaper's sources said that House Democrats are taking up two stopgap funding bills -- one that would fund the government until February 1, and another that would fund it until February 28 -- in their own hail Mary attempt to pressure Senate Republicans to act.

"The strategy is aimed at wavering Republicans who are despondent over the shutdown," the report noted. "But it also faces a tough road ahead, as it's similar to the House strategy from last week of passing bills to reopen the government in a piecemeal fashion."

Last week, the Democrats' bills to reopen government agencies piece-by-piece began drawing defectors -- one attempt garnered eight GOP House members, and another got 12.

Meanwhile, the president's increasingly-frequent public addresses -- including his televised Oval Office address -- appear not to have changed the minds of many Americans in his favor.