Congressional efforts to end the Trump administration policy of splitting up migrant families gained steam Tuesday as bipartisan outrage about the practice mounted.

But Democrats quickly indicated they may not cooperate with a GOP push for a legislative solution, and put the burden on President Donald Trump to change his policy himself.

On Tuesday, Senate Majority Leader Mitch McConnell told reporters that he and "all of the members of the Republican conference support a plan" to keep families together and end the White House practice. The GOP aims to move quickly on a legislative fix, but needs support from at least nine Democrats to get the 60 votes needed to pass one.

"My hope is this is not going to be something we're going to do over a matter of weeks and months but something we can do over a matter of days, hopefully this week," Senate Majority Whip John Cornyn, R-Texas, added.

McConnell said, a bill "would need to be a narrow agreement to fix the problem we all agree needs to be fixed." It means the Senate may not pursue a broader overhaul of the immigration system and border security, as Trump appears to seek.

Numerous Republicans and Democrats in Congress have now called on the White House to immediately halt the policy, which critics have called inhumane and un-American. It is unclear whether a Trump administration that has vehemently defended the policy or a deeply divided Congress can take action to end the widely condemned practice.

But Democrats put the onus on the president to take action himself Tuesday. An animated Senate Minority Leader Chuck Schumer said, "The president alone can fix it with a flick of a pen by signing a presidential order to end the agonizing screams of small children who have been separated from their parents." He shrugged off concerns about the GOP potentially attacking Democrats for not supporting legislation to stop the policy.

"Let's hope we never get to that. Let's hope the president does the right thing and solves the problem, which he can do. That's the simplest, easiest and most likely way this can happen," he said, noting that Congress has tried and failed repeatedly to pass immigration bills.

Many Republicans — who control both chambers of Congress — have said the White House can and should stop the policy on its own.

Sen. Orrin Hatch, the longest-serving Senate Republican and a Trump ally, gathered signatures from 11 other GOP senators on a letter asking Attorney General Jeff Sessions to end the family separations while Congress figures out a faster way to deal with illegal immigrants.

"We support the administration's efforts to enforce our immigration laws, but we cannot support the implementation of a policy that results in the categorical forced separation of minor children from their parents," Hatch said in the letter, which was also sent to Homeland Security Secretary Kirstjen Nielsen on Tuesday.

The Republicans signees include Sens. John McCain, Cory Gardner, Susan Collins, Lisa Murkowski and Bob Corker — all of whom have been willing to challenge Trump either in word or deed at various points in his presidency.

The congressional proposal could mark a compromise of sorts with the White House, as the Trump administration has falsely argued that only Congress can end the policy.