Republicans are expected to try passing several bills to make up for funding gaps caused by the shutdown. Aside from Veterans Affairs, the GOP will also pursue bills to fund national parks and the local Washington, D.C., government.

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These bills are expected to be called up under a suspension of House rules, which would require Democrats to support them if they're going to pass.

"House Republicans rightly acted four times to avoid a government shutdown and protect Americans from the harmful effects of ObamaCare," Salmon said Tuesday. "While Senator Harry ReidHarry ReidFranken emerges as liberal force in hearings GOP eyes new push to break up California court The DC bubble is strangling the DNC MORE continues to refuse to come to the table, this bill makes sure our veterans are taken care of while Congress works to hammer out a final bill."

An aide to Salmon said there are more than 40 co-sponsors of his bill, which was just introduced shortly after noon today.

The bill is similar to legislation offered by Sen. Ted CruzTed CruzTrump's America: Businessmen in, bureaucrats out When Trump says 'Make America Great Again,' he means it Booker is taking orders from corporate pharmaceuticals MORE (R-Texas) in the Senate. Cruz has pushed for Republicans to seek smaller funding bills in response to the shutdown.

Tuesday afternoon, White House spokesman Jay Carney said passing several smaller bills would not be a "serious" way to address the shutdown.