(Updated 4:53 p.m. ) After unions leaders officially rejected the concession package Tuesday, its top negotiator wrote   Gov. Dannel P. Malloy and asked him to “reconvene discussions as soon as possible to see what is necessary to reach an agreement that can be finalized and implemented.”

But Malloy may not welcome another renegotiation session on the $1.6 billion concession package rejected by 4 of the 15 unions. He has repeatedly said he will only clarify the language in the current agreement.

“What’s the sense of negotiating if nothing can ever be approved,” Malloy said during a visit to Mystic Aquarium Tuesday morning. “Rather than prolong the agony it would be better for them to tell us how they can get anything passed.”

Colleen Flanagan, Malloy’s spokeswoman, reiterated Malloy’s feelings on renegotiation Tuesday afternoon.

“The Governor has said all along he’s happy to clarify the language of the agreement if that will allow it to be ratified,” Flanagan said in an emailed statement.

“We wouldn’t characterize what we’ve asked for as really anything more than a reconvening of talks,” Matt O’Connor, SEBAC spokesman, said.

Larry Dorman, another SEBAC spokesman, said he hopes when the governor reads the letter he will agree with what’s in it and decide it is worth continued effort to try and move forward. “We don’t want to see these layoffs, we don’t want to see these service cuts,” he added.

Fifty-seven percent of union membership voted in favor of the package, but due to the 1996 bylaws of the State Employees Bargaining Agent Coalition, 14 of the 15 unions would have had to approve it.

With rejection of the package imminent the legislature gave Malloy increased power to cut the budget last week. His current proposal involves laying off 6,700 state employees, but state agency commissioners are expected to weigh in on the plan before July 8 and could make additional suggestions that call for laying off fewer employees and cutting more services, or programs. Malloy will present the final budget cuts to the Speaker of the House and Senate President Pro Tempore on July 15.

Malloy has said repeatedly he doesn’t want to layoff anyone.

“With a 9.1 percent unemployment rate in the state of Connecticut this is not the time to be laying off a lot of people,” said Malloy. “I don’t want to do it that’s why we negotiated for 60 days to get an agreement, which I thought was a very fair agreement.”

Malloy said it’s now up to the unions to come back and tell him how they expect to get the package passed.

But union spokesman gave no specifics Tuesday of exactly how they planned to pass a package, which will likely become a sticking point for Malloy and his administration.

“We’re in the process of consulting are member leaders and talking about what we can do going forward so that would hopefully alleviate some of the concerns the governor and others have raised,” Dorman said. “Right now it’s really important for us to listen and take note of what our member leaders are telling us, both good and bad, and figure out a way forward.”

On Friday union spokesmen alluded to possibly clarifying the current package, changing SEBAC’s bylaws, and going out to the membership for another vote. O’Connor said the bylaws were looked at by unions leaders, but was unable to share anything more about how they could be changed or even if they’ve presented those changes to Malloy.

“Under the old rules they didn’t pass it but they need to think about how we get from here to there in the future,” Malloy said.

“The collective bargaining process should be allowed to continue,” O’Connor said. “And that was the reason leaders reached out as they did today.”

Dorman said both the unions members who voted for the package and those who voted against it, want it fixed.

“I don’t think anybody in this building wants to see the pink slips issued. I don’t think anybody wants to see services cut,” Dorman said. “We’re hopeful and we’re even optimistic despite the setbacks that have happened we can figure this out and I think the governor and the legislature want to see it figured out too.”

Rep. Diana Urban, D-North Stonington, who accompanied Malloy on his tour of the Mystic Aquarium Tuesday said the thing she most fears is the multiplier effect from the layoffs. She said if there are no major changes in the agreement and it achieves the same amount of savings then she thinks it’s possible the unions could eventually ratify an agreement.

Sen. Andrew Maynard, D-Stonington, who was also on the tour said he thinks the state should do whatever it can “within reason to avoid layoffs,” but ultimately the decision will be made by the governor.

“The public has come to expect us to ratify the agreement that’s there,” Maynard said.

But if Malloy decides to modify the current package he could be seen as someone in the pocket of labor.

House Minority Leader Lawrence Cafero, R-Norwalk, said in a phone interview that he has to wonder who is running the state. The legislature and the governor, or the labor unions.

“Whose running the show here?” Cafero said.

He said the state shouldn’t sit back and let the unions do what they want to, then react.