(CNN) Virginia Democrat Sen. Tim Kaine and Arizona Republican Sen. Jeff Flake are trying once again to get Congress to debate and vote on authorizing military action against ISIS.

The senators on Thursday unveiled a new Authorization for Use of Military Force against ISIS, al Qaeda and the Taliban, their second attempt to prod their colleagues to formally vote on the military effort.

But their first attempt, introduced in 2015, went nowhere . Congressional leadership has shown no interest in taking up a new war authorization, as both the Obama administration and now the Trump administration have said they have the legally authority needed under the 2001 authorization passed after the September 11, 2001 attacks, and the 2002 Iraq War authorization.

Lawmakers are wary of taking a formal vote on the issue, cognizant of how then-Sen. Hillary Clinton's 2002 Iraq War vote was used against her in both the 2008 and 2016 presidential elections

"I frankly think this is, partly, hard because people don't want to cast a war vote," Kaine said. "Because there's going to be a consequence. It should be the gravest vote we ever cast. People want to duck from that if they can."

But the two senators said they were hopeful that their bipartisan effort can finally gain some steam. They said Senate foreign relations chairman Bob Corker indicated Thursday he was interested in taking some action on the bill in his committee in the near future.

Flake said that members do want to support a war vote in theory, but getting into the details of what a war authorization looks like has been the biggest sticking point.

"You'd be hard-pressed to find a member of Congress who has said, 'no, we shouldn't do an AUMF,'" Flake said. "It's always, 'we should do it,' but people have had a hard time coming up with something (to) fit the evolving threat that we've had."

Critics of the US war against ISIS say that the Obama and Trump administrations are relying on an overly broad war authorization that has no restrictions on timing or location, and a broad definition of who could be targeted.

"When I voted in 2001 to authorize military force against the perpetrators of the September 11 attacks, I had no idea I would be authorizing armed conflict for more than 15 years and counting," said Flake, who was a House member then.

Kaine and Flake's resolution would repeal both the 2001 and 2002 war authorizations, putting in place a new authority that would last for five years.

The resolution would authorize US military operations against ISIS, al Qaeda, the Taliban as well as "associated" forces, which the measure would require the administration to define in a report to Congress.

The resolution would allow the military to conduct operations around the globe — just like the 2001 war authorization — but requires the president to tell Congress why operations are being conducted if they are beyond Afghanistan, Syria, Iraq, Yemen, Libya and Somalia.

The small bipartisan cadre of lawmakers who have called for Congress vote on the ISIS fight said they were hopeful the Trump administration would provide a new avenue for Congress to take up the issue.

"We had a bill in for a couple of years nobody was interested in -- we tried to wordsmith differences between Democrats and Republicans -- so we're actually exploring some different ways of coming at it," Kaine said in March of the measure he's now introducing Thursday.

President Donald Trump's decision to launch missile strikes in Syria also sparked a slew of calls for a war authorization against both ISIS and the Bashar al-Assad regime, but it did not lead to any movement on the issue.

Kaine and California Rep. Adam Schiff, who has his own ISIS war authorization bill, have called on the President to justify his strikes in Syria, but both authorization measures are focused on ISIS and don't deal with military action against the Syrian government.

Corker, a Tennessee Republican whose committee would handle a new war authorization, has said he wants to see a broader ISIS strategy from the Trump administration before considering congressional authorization.