The two leaders struck a tone of bipartisanship in a joint interview with CNN's Anderson Cooper Thursday night at the Congressional Baseball Game, though they stressed they still had their policy differences.
Those differences were amplified Friday when Schumer sent a letter Friday requesting an all-senators meeting in the Old Senate Chamber next week.
Senate Democrats have been blasting Republicans for deliberating the health care bill behind closed doors, saying they've only been learning of the ongoing negotiations through media reports.
"Please accept our invitation to sit down together in the old Senate Chamber so we can hear your plans and discuss how to make health care more affordable and accessible in the United States," Schumer writes in the letter to Senate Majority Leader Mitch McConnell, which was sent to the press by Schumer's office.
The Senate stopped meeting in the Old Senate Chamber in 1859, and the room is now occasionally used for ceremonial events or moments of bipartisan negotiations. In 2013, for example, the body met there as it debated changes to filibuster rules.
Republicans have been working on a Senate version to repeal and replace Obamacare since House Republicans passed a measure more than a month ago. They contest accusations of secrecy, arguing that they've been holding open debates and hearings on their disagreements with Obamacare since it became law seven years ago.
While Republican leaders in the Senate, like McConnell, have pitched a vote before the July 4 recess, many in the party have expressed doubt
that Republicans will reach that deadline with only 10 legislative days left.
Meanwhile, Democrats have been holding news conferences and calling on Republicans to share with them more details of the negotiations. Sen. Ron Wyden, D-Oregon, said Thursday that the party plans to increase publicity efforts to call out the GOP on the issue.
"I believe you're going to see a grassroots juggernaut from one side of the country to another that the next two weeks are pivotal," he told reporters. "We believe this is go time."
David Popp, a spokesman for McConnell, argued that the letter suggests Democrats are now open to the "repeal" component of the "repeal and replace" health care effort. He alluded to Schumer's comments earlier this year in which he said Democrats will work with Republicans on repairing Obamacare as long as Republicans back off of the "repeal" part of it.
While Schumer's letter does not say Democrats are now open to repealing Obamacare, Popp argued that the letter's premise of wanting to meet suggests that Democrats have dropped the precondition.
He also pointed to a line in the letter in which Schumer said Democrats would like to "discuss how to make health care more affordable and accessible."
"They've also acknowledged in yesterday's letter that Obamacare is not affordable or accessible. That admission is another positive step from a party that, until now, has been wedded to the unsustainable status quo and its consequences for families across the country," he said, urging Democrats to "propose ideas" to solve existing problems with Obamacare.
Popp did not say whether he thinks members of the two parties will actually meet in the Old Senate Chamber.
A spokesman for Schumer, Matt House, confirmed that Democrats are not giving up on their push for Republicans to abandon the "repeal" component.
"We're asking for this meeting because the Republicans have refused to make their bill public or hold hearings on it. We hope that this meeting will lead to Republicans abandoning both repeal and their repeated sabotage of our health care system, which has sowed uncertainty in the marketplaces," he said in a statement.