House and Senate negotiators are closing in on a final $320 billion-plus omnibus fiscal 2019 package, after what Senate Appropriations Chairman Richard C. Shelby called a “productive” and “positive” meeting with President Donald Trump on Thursday.

“[The president] said to me again he would like for us to wrap it up, to get a legislative solution,” the Alabama Republican told reporters at the Capitol after the meeting. “We’re negotiating on the substance, serious stuff now. …This is the most positive I’ve been or I’ve seen in the talks since, oh gosh, maybe ever.”

One of the concerns expressed in recent days by Democratic leaders is Trump’s penchant for doing the unexpected, overruling his top aides and reversing what GOP leaders thought he had already decided. But Shelby sought to tamp down such fears after talking with the president.

“If we can work within some of the parameters we talked about today — we’ll keep to ourselves right now — I think he would sign it,” Shelby said.

Trump, speaking to reporters at the White House, did not downplay the possibility of a deal. “We’ll see what happens but I certainly hear they’re working on something, and both sides are moving along,” Trump said. 

Senate Homeland Security Appropriations ranking member Jon Tester, D-Mont., told reporters Thursday that a conference committee agreement could come as soon as Friday. “I think it’s entirely possible we could have a deal in a timely manner, which would be tomorrow,” Tester said.

Tester’s Democratic counterpart in the House, Rep. Lucille Roybal-Allard, D-Calif., was more cautious, however. “We’re waiting for a counteroffer right now,” she said. “Nothing has been decided and nothing is finalized.”

Shelby said he hoped the 17 members of the conference committee on the Homeland Security bill, which is the expected vehicle for the other six outstanding spending measures, could meet to bless the package on Monday. That would allow for legislative text to be released in time to get the conference report to the House floor by Thursday, in keeping with the chamber’s 72-hour rule, followed by Senate action.

“I think we’re looking at Monday right now,” Shelby said. “But there’s a lot to do. We’ve got some sharp people working on this on both sides. But let’s see if we can get there.”

When asked what the next steps are, Shelby replied: “We hope we’ve got a deal. If we haven’t got a deal, we probably won’t get a deal.”

The deadline is next Friday, Feb. 15, when the latest stopgap measure expires.

House and Senate negotiators have been attempting to work within the DHS bill’s regular discretionary spending cap of roughly $49 billion, while providing a robust mix of border security funding — some “enhanced barriers,” as well as more technology to root out narcotics trafficking plus additional border and law enforcement personnel. The other challenge has been avoiding cuts to other DHS agencies such as the Transportation Security Administration and Coast Guard.

The Senate Republican approach, backed by the White House, was to designate nearly $5.6 billion in emergency funds to bolster Trump’s demand for 234 miles of steel barriers along the border. Democratic conferees rejected the add-ons and said Wednesday there was agreement to live within the regular budget caps.

Shelby and Senate Appropriations ranking member Patrick J. Leahy, D-Vt., said the remaining six spending bills for fiscal 2019 — Agriculture, Commerce-Justice-Science, Financial Services, Interior-Environment, State-Foreign Operations and Transportation-HUD — were largely completed.

Both chambers last month released versions with nearly identical text and funding levels, with the exception of Financial Services, which had roughly $400 million more directed towards improvements at land ports of entry in the House version. House Democrats offered a reduced allocation for Homeland Security to make up the difference.

“I think we’re close on all the bills,” Leahy said.

Paul M. Krawzak and Lindsey McPherson contributed to this report.