(Adds detail of Senate Republican health plan)
WASHINGTON, June 6 (Reuters) - President Donald Trump met with Republican congressional leaders on Tuesday to rejuvenate efforts to overhaul the U.S. healthcare and tax systems as investigations into Russian meddling in the 2016 election campaign weigh on his administration.
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, such as Senator Orrin Hatch, said they may vote by early July.
Republicans have little margin for error as they hold a two-seat majority in the 100-seat chamber and remain divided over key questions such as whether to scale back the Medicaid health plan for the poor.
"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.
Republicans have sought to overturn Democratic former President Barack Obama's signature domestic healthcare law since it was enacted in 2010.
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. (Additional reporting by Ayesha Rascoe, Richard Cowan, Mohammed Zargham, Roberta Rampton, Amanda Becker and Susan Heavey; Writing by Andy Sullivan and Alistair Bell; Editing by Simon Cameron-Moore and Frances Kerry)