But his continuing push for a government-run plan is generating fierce opposition.
"If we go to a government plan, the government will be setting prices. They'll be telling people what they can do. They'll be injecting themselves between your doctor and yourself," said Sen. Orrin Hatch, R-Utah, calling a public plan "tremendously costly."
Hatch and others argue that the government should not create a new public health care option when it can't pay for the ones it already has.
Just one part of Medicare, for instance, has promised some $36 trillion more in benefits than the government projects in revenue.
Medicare and Medicaid, health care for the poor, are in such dire straits that Obama said in his radio address that Americans are right to worry that the "ballooning costs" of both could lead to a "fiscal catastrophe down the road."
But the president insists that any new plan will be paid for, and that it will not add to the deficit.
Republicans are suspicious, in part because Obama has been working only with Democrats, even meeting with them exclusively just before leaving on his overseas trip.
Meanwhile, some Democrats want to add to any reforms, with Sen. Tom Harkin, D-Iowa, insisting Monday on as much as $4 billion more each year for the disabled.
"I know that's a lot of money, but we're talking about a one-trillion-dollar health reform bill," Harkin said.
As Democrats argue behind the scenes about what to do, Obama is trying to pressure Congress to act quickly.
But critics note there is no bill on the table -- nothing to vote on or even debate.
"Pres Obama you got nerve while u sightseeing in Paris to tell us 'time to deliver' on health care. We still on skedul/even workin WKEND," he said in his online "tweet," or short-form text message.
And when the president told his supporters a few days ago that it was now or never for health care, House Republicans cringed.
"It's not our fault that they didn't have a (health secretary) until the end of April. It's not our fault that it is June and there is still not a bill. To not have a bill in June and say it is now or never is not fair to the American people," said Rep. Roy Blunt, R-Mo.
Some Republicans and Democrats agree this has to be a bipartisan effort.
Hatch warned that if the Democrats go the partisan route, they know they're going to "eat" the consequences for the "rest of their lives"