Former Democratic National Committee chairwoman Donna Brazile told Fox News Thursday that House Speaker Nancy Pelosi, D-Calif., would soon put forward "an alternative package" in a bid to end the ongoing partial government shutdown.

Brazile told "The Daily Briefing with Dana Perino" that she did not know exactly when the measure would be announced, but said it would contain "a lot of border security items."

"I think what you're seeing now in the Senate, you'll see it in the House soon," Brazile said. "We want to get this government shutdown over."

Earlier Thursday, Pelosi denied reports that House Democrats were working on a counteroffer to President Trump's demand for $5.7 billion to fund his long-promised wall along the U.S.-Mexico border.

"That's not true." Pelosi told reporters at her weekly news conference. "That's not true. That's not true."

"We are doing what we have been doing all along," the speaker added. "We have been working on our congressional responsibility to write bills – appropriations bills to keep government open. Many of those bills have come to the floor again and again just this week."

Democratic leaders have repeatedly insisted that the government fully reopen before any negotiations on a border barrier can begin.

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Earlier Thursday, the House passed the latest in a series of measures aimed at reopening the government, with a 231-180 vote to open the Homeland Security Department. It was the 11th attempt to pass a bill ending the shutdown that began over a month ago.

On the Senate side, Republican and Democratic proposals to open the government failed procedural votes. The Republican version would have provided money for the wall and temporarily shielded from deportation 700,000 young immigrants who came to the U.S. illegally as children — a protection Trump has tried terminating. The Democratic proposal would have reopened federal agency doors through Feb. 8 while congressional negotiators seek a budget accord.

The Associated Press contributed to this report.