Senate Republicans on Tuesday moved to end a political crisis inflicted on the party by President Trump Donald John TrumpBiden on Trump's refusal to commit to peaceful transfer of power: 'What country are we in?' Romney: 'Unthinkable and unacceptable' to not commit to peaceful transition of power Two Louisville police officers shot amid Breonna Taylor grand jury protests MORE by voicing support for an immigration bill that would end the separation of children from their parents and guardians at the border.

Senate Majority Leader Mitch McConnell Addison (Mitch) Mitchell McConnellFEC flags McConnell campaign over suspected accounting errors Poll: 59 percent think president elected in November should name next Supreme Court justice Mark Kelly: Arizona Senate race winner should be sworn in 'promptly' MORE (R-Ky.) said Republicans need to fix the problem through legislation, and that the fix should be narrowly tailored.

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“My assumption is in order to fix this problem you can’t fix all the problems,” he told reporters.

He also noted pointedly that “all of the members of the Republican conference” support a plan that would keep children with their families, underlining the strong opposition among GOP senators to Trump’s “zero tolerance” policy, which has resulted in children being taken from their parents.

“We need to fix the problem and it requires a legislative solution,” McConnell said during his regular Tuesday afternoon appearance before reporters — and just an hour or so after a defiant Trump showed no signs of backing down during a fiery speech.

Lawmakers in both parties, along with conservatives in the media and business, have panned the Trump policy for separating families, which has created campaign fodder for Democrats.

The Wall Street Journal’s editorial board on Tuesday said the policy risked losing Republicans their majorities in the House and Senate, underscoring how the crisis has unnerved conservatives ahead of the midterm elections.

McConnell’s support for a narrow bill is telling given work by House Republicans on a broader measure.

The legislation in the House, a compromise between GOP centrists and conservatives, would stop the Department of Homeland Security from separating parents from their children. But it also includes border security measures, funding for Trump’s wall on the border and language that would allow “Dreamers” who came to the United States illegally as children to remain in the country.

It faces a difficult road in the House, and would almost certainly be dead on arrival in the Senate, where it would need support from Democrats to overcome a filibuster.

The House is expected to vote on the compromise legislation and a more hard-line bill on Thursday.

If the bills fail to pass, it could give leverage to a narrow Senate bill, though that legislation would also face questions.

Democrats argue, accurately, that Trump could end the separation of families by giving an order from the White House.

While Trump has blamed Democrats for the issue, Attorney General Jeff Sessions Jefferson (Jeff) Beauregard SessionsGOP set to release controversial Biden report Trump's policies on refugees are as simple as ABCs Ocasio-Cortez, Velázquez call for convention to decide Puerto Rico status MORE introduced the policy that led to the separation of families this spring. Previous administrations under former Presidents George W. Bush and Obama chose not to take the step of splitting parents from their children to deter illegal border crossings.

Democrats say the onus is on Trump to prevent immigrant children from being separated from their parents.

“There are so many obstacles to legislation,” Senate Democratic Leader Charles Schumer Chuck SchumerCruz blocks amended resolution honoring Ginsburg over language about her dying wish Senate Democrats introduce legislation to probe politicization of pandemic response Schumer interrupted during live briefing by heckler: 'Stop lying to the people' MORE (N.Y.) told reporters Tuesday. “Legislation is not the way to go here when it’s so easy for the president to sign it.”

He argued any legislative fix would quickly become a magnet for “unacceptable additions,” such as proposals to toughen enforcement provisions.

But Schumer did not completely rule out Democratic support for a narrow bill.

Asked if his remarks meant Democrats would not support the bill favored by McConnell to keep immigrant families together while seeking asylum on the U.S. border, Schumer emphasized “the president can change it with his pen.”

Senate Republicans are discussing a bill being crafted by Republican Whip John Cornyn John CornynThe Hill's Campaign Report: GOP set to ask SCOTUS to limit mail-in voting Liberal super PAC launches ads targeting vulnerable GOP senators over SCOTUS fight Senate GOP faces pivotal moment on pick for Supreme Court MORE (Texas) that would require families be kept together in a “humane setting” while waiting for a hearing with an immigration judge.

The proposal would also expedite hearings for those families to minimize their time in detention.

Other Republicans, such as Sens. Ted Cruz Rafael (Ted) Edward CruzCruz blocks amended resolution honoring Ginsburg over language about her dying wish Trump argues full Supreme Court needed to settle potential election disputes Press: Notorious RBG vs Notorious GOP MORE (Texas) and Tom Cotton Tom Bryant CottonGOP brushes back charges of hypocrisy in Supreme Court fight Trump uses bin Laden raid to attack Biden Battle lines drawn on precedent in Supreme Court fight MORE (Ark.), have their own proposals.

Cruz plans to introduce a bill this week that would double the number of federal immigration judges, authorize new shelters to keep families together and require the Department of Homeland Security to keep children with parents unless there are aggravated criminal charges.

Cotton wants to add language to an appropriations bill that would overturn the 1997 Flores settlement, which requires border agents to release children from detention after 20 days, and provide money for family housing units.

The more that is added to a bill, however, the more difficulty it could have picking up Democrats. And there’s no guarantee Democrats would back a bill that simply overturns Trump’s policy, though the chances could grow with the defeat of the House measures.

McConnell was careful to avoid criticizing Trump in his Tuesday remarks. He declined to say whether he spoke personally to Trump about the controversy or offer his personal thoughts on whether the administration should “pause” its policy, as Sen. Orrin Hatch Orrin Grant HatchBottom line Bottom line Senate GOP divided over whether they'd fill Supreme Court vacancy MORE (R-Utah) requested in a Tuesday letter to Sessions.

Other Republicans criticized Trump’s policy more directly.

“While I firmly support enforcing our immigration laws, I am against using parental separation as a deterrent to illegal immigration,” Sen. Pat Roberts Charles (Pat) Patrick RobertsThe Hill's Morning Report - Sponsored by Facebook - Washington on edge amid SCOTUS vacancy The Hill's Morning Report - Sponsored by National Industries for the Blind - Trump seeks to flip 'Rage' narrative; Dems block COVID-19 bill GOP senators say coronavirus deal dead until after election MORE (Kan.), a senior Republican, tweeted on Monday.

In a speech to the National Federation of Independent Business, Trump was full of fury about the border. He blamed the “fake news” media for stirring up the controversy, saying it was aiding smugglers and human traffickers. He also said Mexico was not doing enough to stop illegal immigration.

The president said he wanted Congress to grant him “the legal authority to detain and properly remove families together as a unit,” and questioned the need for more judges to handle immigration cases, something supported by lawmakers in both parties and his own Justice Department.

“I don’t want judges,” he said. “I want border security. I don’t want to try people. I don’t want people coming in.”

Juliegrace Brufke and Jordain Carney contributed.