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 ReidOPINION | 5 ways Democrats can win back power in the states THE MEMO: Trump's base cheers attacks on McConnell It's time for McConnell to fight with Trump instead of against him 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 CruzThe media couldn't be more blatant in distorting Trump's words on Charlottesville Curtis wins GOP primary for House seat vacated by Jason Chaffetz Kimmel: Let’s make Trump a king so he has no power 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.