"We're also looking at the likelihood of another bill ... with some anomalies, some additions that takes us to Jan. 19 ," he told reporters.

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The government is scheduled to shut down Saturday unless Congress passes a bill to fund the government by the end of Friday.

House Republicans are poised to move forward with a bill that links a full year of defense funding with a continuing resolution to fund the rest of the government for a few weeks.

But that bill can't pass the Senate where GOP leadership needs the support of at least eight Democrats or independents to help pass legislation, if they can hold their entire caucus together.

Forty-four of the Democratic caucus's 48 members have warned they will not support the House's legislation, meaning it will fail to get the 60 votes needed to break a procedural hurdle.

Any changes made by the Senate will require the funding bill to bounce back to the House for a second vote.

The Senate is weighing dropping a myriad of issues into its funding bill, including two health-care provisions, an extension of a controversial surveillance program and disaster relief funding.

"I talked to Leader [Kevin] McCarthy [R-Calif.] and they are going to send us over a bill they can pass. And then we'll take it up and decide what we can pass," he said.

Lawmakers from states hit by a spate of recent natural disasters are also pushing for supplemental funding to be included in the short-term funding bill.

Cornyn noted lawmakers "were still trying to figure out" how disaster relief shakes out.

"[We'll see] how much they can get done on the House side, and then we'll take a look at that and see if we need to do more," he said.

McConnell noted that Congress also needed to address the Children's Health Insurance Program (CHIP), a key veterans program and waiving the Pay-As-You-Go Act for the GOP tax plan, which could otherwise spark cuts to Medicare.

Democrats are also pushing for a deal on the Deferred Action for Childhood Arrivals (DACA) program, which the Trump administration announced earlier this year that it was ending.

Senators are locked in behind-the-scenes negotiations to pair a DACA fix with a border security package, though it appears unlikely they would be able to get a deal this year.