WASHINGTON (Reuters) - President Donald Trump pressed Republican congressional leaders on Tuesday to complete their overhaul of the U.S. healthcare system as lawmakers said they were making progress on a contentious effort that threatens to overwhelm their legislative agenda.

In a White House meeting, Trump welcomed indications by senior Republicans that the Senate might vote on a healthcare bill in July, before it breaks for the summer, after the House of Representatives passed its own version in May.

“Now the Senate I’m sure will follow suit and get a bill across the finish line this summer that will be great healthcare and I’m looking forward to seeing it,” Trump said.

The gathering included Senate Majority Leader Mitch McConnell, and his No. 2, Senator John Cornyn, along with House Speaker Paul Ryan and his deputies, Representatives Kevin McCarthy and Steve Scalise.

Buffeted by criticism on many fronts, Trump wants faster action from his fellow Republicans who control Congress, pressing lawmakers to finish the job of dismantling the Obamacare healthcare law and move on quickly to another of his priorities: tax cuts.

McConnell said Senate Republicans are “getting close” to a healthcare plan after he presented an outline at a lunchtime meeting but he declined to say when he might bring it up for a vote. Other senior Republicans say they hope to vote by early July.

“The leader would like to do this soon,” Senator Orrin Hatch told reporters.

The Senate requires a 60-vote supermajority to advance most legislation but Republican Mike Enzi, head of the Senate Budget Committee, said the health bill that passed the House qualifies for an expedited process that would require only 51 votes.

Still, Republicans have little margin for error because they only control 52 seats in the 100-seat chamber and remain divided over key questions such as how quickly to scale back the Medicaid health plan for the poor.

U.S. President Donald Trump walks out to annouce proposed reforms to the U.S. air traffic control system during an event at the White House in Washington, DC, U.S. June 5, 2017. REUTERS/Jonathan Ernst

“You have to get 50 people - that means they have to get me and a lot of people who are concerned about Medicaid,” said Senator Shelly Moore Capito of West Virginia, a state that depends heavily on the program.

Medicaid was expanded under Obamacare but the House bill would phase that expansion out in 2020. The Senate bill might keep the expansion in place beyond that date, said Republican Senator John Barrasso.

COMEY TESTIMONY

Whatever the White House’s efforts to push ahead with policy plans, there will be a spotlight on testimony by James Comey, the FBI director fired by Trump last month, to the Senate Intelligence Committee on Thursday.

Senators will question Comey on whether Trump tried to get him to back off an FBI investigation into ties between the president’s 2016 campaign and Russia, an attempt that critics have said could constitute obstruction of justice. Trump denies any collusion with Russia and has called the investigation a “witch hunt.”

The Justice Department has appointed a special counsel to oversee its probe into the Russia issue and several congressional panels also are investigating the matter.

Elected on pledges to overhaul the healthcare system and slash taxes, Trump has yet to achieve a major legislative win, and time is running out before lawmakers leave Washington for the August break.

There has been little progress on healthcare since the House passed its bill. McConnell told Reuters late last month he did not yet know how to amass the votes needed to pass a bill on healthcare.

He appeared to make some progress on Tuesday. Some Republican lawmakers praised an outline that McConnell presented at a lunchtime meeting, although they cautioned that much work remained to be done.

Senator Bill Cassidy said the plan offered more protections than the House bill for people who already have pre-existing illnesses when they apply for insurance coverage - a major point of contention.

“I think personally they are moving in the right direction,” he said.

The House healthcare bill could result in 23 million people losing insurance, according to the nonpartisan Congressional Budget Office. The bill also would reduce federal deficits by $119 billion between 2017 and 2026, according to the analysis.

Congress might then turn its focus to overhauling the tax code in September. While the administration would prefer that such changes not add to the national debt, Marc Short, Trump’s top aide on Capitol Hill, told reporters on Monday that the top priority would be cutting taxes.

The Trump administration has outlined a broad plan that would cut tax rates for businesses and streamline the tax system for individuals. The proposal has been short on details, including how much the tax cuts would cost and what loopholes would be closed.

Others are eager to move past healthcare as well.

“We need to bring this to an end and move to taxes,” said Republican Senator Lindsey Graham.