(Adds Cornyn, Thune quotes)
WASHINGTON, June 6 (Reuters) - President Donald Trump is meeting congressional leaders on Tuesday to push for efforts to overhaul the U.S. healthcare and tax systems, as investigations into Russian meddling in the 2016 election campaign weigh on his administration.
Senior Republicans suggested the Senate might vote on a healthcare bill in July, before it breaks for the summer.
Trump wants faster action from his fellow Republicans in Congress to advance his legislative agenda, pressing lawmakers to finish the job of dismantling the Obamacare healthcare law and move on quickly to another of his priorites: tax cuts.
But Republican lawmakers remain divided on healthcare after the House of Representatives passed a bill to replace Obamacare in May.
The party's senators will discuss options for the legislation at a weekly policy luncheon on Tuesday, said John Cornyn, the No.2 Republican in the Senate.
There could be a vote in the Senate on healthcare in July sometime, he said.
"Weve been talking about this for seven years. Nows the time to start coming up with some tangible alternatives and building consensus, Cornyn said on Monday night. Republicans have sought to overturn Democratic former President Barack Obama's signature domestic 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 Trump fired 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 are also 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.
"Big meeting today with Republican leadership concerning Tax Cuts and Healthcare. We are all pushing hard - must get it right!" Trump wrote on Twitter early on Tuesday.
Trump will hold talks at the White House at 3 p.m. (1900 GMT) with leaders including House of Representatives Speaker Paul Ryan, Senate Majority Leader Mitch McConnell and Cornyn.
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.
Plans to roll back the government's Medicaid health plan for the poor and disabled are still a big sticking point among the congressional Republicans, said Senator John Thune.
"I think theres a way we can get there, none of this is going to be easy, he said on Monday.
The House healthcare bill could result in 23 million people losing insurance, the Congressional Budget Office estimated, although Republicans have challenged that conclusion. The bill would also reduce federal deficits by $119 billion between 2017 and 2026, according to the analysis.
The White House also wants to see a vote to raise the government's borrowing authority, known as the debt limit, before the congressional summer break.
Republican fiscal conservatives routinely demand budget cuts and other concessions as a price for raising the debt limit, setting up a likely fight.
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.
"We want it to be revenue neutral, and we are still supportive of tax reform, but I am also saying to you that what we believe is most important to get the economy going is the tax cuts," he said.
The Trump administration has outlined a broad plan that would cut tax rates for businesses and streamline the tax system for individuals. But, the proposal has been short on details -- including the cost of the tax cuts and what loopholes would be closed. (Additional reporting Richard Cowan, Susan Cornwell, Roberta Rampton and Susan Heavey; Writing by Alistair Bell; Editing by Simon Cameron-Moore and Frances Kerry)