"I don't think there are any announcements to be made but there some very serious conversations," said Sen. John Cornyn of Texas, the second-ranking Senate Republican.
Aides from each party said senators were narrowing in on a roughly $1 billion package to respond to the developing public health crisis. The White House had asked Congress for $1.9 billion but were initially rebuffed by GOP lawmakers who didn't want to approve any new funds right away. Republicans argued that hundreds of millions of dollars leftover from another public health challenge -- the battle against Ebola -- could be repurposed for Zika, a plan the White House eventually agreed to.
But Republicans have softened their opposition to new funding in recent days in the face of mounting pressure from Democrats and more reports about the potential spread of the virus to more areas in North America.
"We recognize that there is an urgency associated with the threat," Appropriations Committee Chairman Thad Cochran, a Republican from Mississippi, announced at a hearing of his committee.
"We are working on it," said Sen. Patty Murray of Washington, a principal negotiator for the Democrats. "I'm not going to discuss the negations but I can tell you the Republicans have come a long way. They are working in good faith and we are looking for a way to get it done in a timely fashion."
In a sign of their high interest in the virus, Dr. Anthony Fauci, the head of infectious diseases at the National Institutes of Health, briefed Senate Democrats on the issue Wednesday.
"It could be a good deal of money," acknowledged Sen. Richard Shelby of Alabama, a senior Republican member of the Appropriations Committee. "But it could be one awful challenge for all of us," he added about the potential spread of the unpredictable virus.
One key sticking point is whether the money should be considered emergency funding so the costs would not have to be offset and would not count towards the caps of last year's budget deal. Republicans want those caps to apply to the money while Congressional Democrats and the White House don't.
"It's absolutely critical," Murray said.
In the House, Democratic Leader Nancy Pelosi echoed that view.
"Zika should an emergency supplemental," Pelosi told reporters. "There is always a recognition if there is an emergency, it would be outside the caps."
Cochran said he planned to attach the spending request to one of the existing appropriations bills the Senate is working on this spring, complicating Democrats' preference to make it a standalone emergency bill.
Republicans said they were moving forward with President Barack Obama's request even as they complained the White House still had not provided full answers to their questions about how exactly the money would be used.
"The administration continues to delay response efforts by refusing to provide basic budgetary information to Congress on their Zika funding request," said Rep. Hal Rogers, a Republican from Kentucky and chairman of the House Appropriations Committee. "What is needed right now ... to fight the disease?"
Without the answers he's wants from the administration, Rogers has instructed his committee staff to work on its own determination about what it would cost in the short term -- the next five to six months -- to respond to the Zika threat with work on a vaccine and other needs, according to a senior House committee source.
"If necessary, we are prepared to act," the source told CNN.
But the talks are on two separate tracks in the House and Senate, with the focus in the House on looking at different options other than adding to a spending bill for 2017. It's unclear if legislation would be a separate supplemental spending bill or attached to another legislative vehicle.
Pelosi is also pushing hard to add money to address the water crisis in Flint, Michigan, to any bill with Zika funding, something that is a non-starter for most House Republicans.
House Speaker Paul Ryan assured reporters the GOP takes Zika "extremely seriously."
"We see our job as good stewards of the taxpayer dollars, as making sure that we have answers to our questions about how we fund these things," he said. "But this is something we're going to get done to make sure that we're on top of this problem."
Senate Majority Leader Mitch McConnell suggested he was open to the White House request at a news conference Tuesday.
"Look, it's an important issue. We're in discussion with them now about much they really need," he said. "I don't think that in the end there will be any opposition to addressing what we think is going to be a fairly significant public health crisis."
Sen. Dick Durbin of Illinois, the second-ranking Senate Democrat, said Republicans came around to new funding after realizing how serious the spread of the virus could be.
"They don't want to get caught on the wrong side of history when it comes to Zika," he told CNN.