Reps. Jim McGovern, D-Massachusetts, and Tom Cole, R-Oklahoma, are circulating a letter to House Speaker Paul Ryan, which has not yet been released publicly, calling for the House to debate a war authorization when Congress returns from its two-week recess later this month.
And House Democrats held a conference call earlier this week to discuss their own strategy on a war authorization, in which Congress would vote to give Trump the authority to conduct military action but potentially limit the steps he could take, such as deploying ground combat troops.
They plan to press Trump to explain whether he will conduct future attacks against Syrian President Bashar al-Assad's regime as well as push for Congress to reassert its constitutional authority to declare war, congressional aides and lawmakers on the call told CNN.
The call follows a letter last week from House Minority Leader Nancy Pelosi that called for Congress to return from recess to debate Trump's military action in Syria, after Trump authorized a strike of 60 Tomahawk missiles against a Syrian government facility in response to a chemical weapons attack that the US has pinned on Assad.
"I just don't get a good feeling that whatever we're doing has been thoroughly thought out and there's some kind of a plan," said New York Rep. Eliot Engel, the top Democrat on the House Foreign Affairs Committee. "He needs to tell us what his strategy is in Syria, what he thinks he's going to do in the future -- and whatever he decides to do, the President has to come to Congress for authorization."
But there are plenty of signs that Congress is no more likely to take up a war vote with Trump than it was under President Barack Obama, who went to Congress in 2015 with an ISIS war authorization that went nowhere.
Many lawmakers remain wary of the political fallout should the war become unpopular or go badly -- Hillary Clinton's Iraq War vote still looms in the minds of many on Capitol Hill -- and there's disagreement over what the authorization should establish: restrictions on the commander in chief or a freer hand.
The Trump administration has said it already had the authority to conduct last week's missile strikes in Syria in response to the chemical weapons attack there, which gives Republican leaders political cover to argue that a vote on the military action is not necessary.
"The leadership of neither party so far has wanted to do this," Cole said. "I respect that. They don't want to expose their members to what they think could be a tough or controversial vote. But you get elected to vote, and you do have a constitutional responsibility."
A House Republican aide said it was hard to see a war authorization, formally known as an Authorization for Use of Military Force, moving forward at this point, given that the White House has said it's not planning to go into Syria.
Trump said in a Fox Business interview this week, "We're not going into Syria." But he and other top administration officials have suggested they would respond to future chemical attacks.
After last week's strikes, Ryan said the US military response was "fully within the President's authority." He added it was "appropriate" for the administration to consult with Congress on its next steps but did not mention a war authorization.
The Constitution states that Congress is the branch of government that should declare war, not the executive branch. But the Constitution also provides the President with authority to use the military as needed as the commander in chief, CNN legal analyst and national security law professor Steve Vladeck noted
McGovern, who has been a critic of military action in Iraq and Syria under Obama and Trump, told CNN he was prepared to use procedural tactics in the House to force a vote on Trump's military actions if Congress has not received a Syria strategy from the White House.
"We fully expect the President will deliver to Congress an AUMF that he wants us to debate and vote on," McGovern said. "If he doesn't, then we're going to have to think about what our options are -- one of them may be bringing a privileged resolution to withdraw US military forces by a given date from Syria."
While the missile strikes against Assad have sparked new calls for a war vote, some lawmakers say the focus should still be on an authorization against ISIS, especially if the strikes on the Assad regime were just a one-time response to the chemical weapons attack.
"The Syrian strikes last week hasten the interest, but what I am working on with my colleagues is not with respect to action in Syria," said Sen. Tim Kaine, a Virginia Democrat who is working with Arizona Republican Sens. Jeff Flake and John McCain to try to craft a bipartisan war authorization.
The fate of a war measure may ultimately boil down to whether Trump's strike against Assad last week was the beginning of a larger campaign against the Syrian leader.
House Foreign Affairs Chairman Ed Royce would be the point person in the House for a war authorization. The California Republican told CNN's Wolf Blitzer this week that he believes authorization would be necessary only if the Trump administration takes military action against Assad for a duration longer than 60 days, the amount of time the executive branch is provided to take military actions without congressional approval in the War Powers Act.
"If this goes out more than 60 days, of course, there needs to be an authorization of use of military force in order to take out any other gas attack," Royce said.