NEW BRITAIN — Gov. Dannel P. Malloy took a small step Wednesday toward renewed talks about concessions with unionized state employees when he said he likely would send an aide to inquire how labor leaders hope to revise their contract amendment ratification rules.

But the governor also warned that without clear direction first from labor about how a difficult ratification  process might be reformed, there is little for the two sides to talk about. Nearly 60 percent of participating union members voted in favor of concessions last month, but ratification still failed.

Malloy also said his administration remains focused on meeting over $700 million in savings targets needed to balance the new fiscal year’s $20.14 billion budget, and that includes continuing to plan for an estimated 6,600-plus layoffs.

“Maybe what I should do is send someone to discuss what the clearer road should be,” Malloy said, speaking to reporters about concessions after promoting one of his jobs creation programs at the Main Street construction site of the new city police station.

The unions’ chief negotiator, Daniel Livingston, sent Malloy a letter Tuesday asking that labor and the administration reconvene discussions.

After a tentative concession package projected to save $1.6 billion over two years failed in a union vote last month, Malloy said he wouldn’t negotiate a new deal, but did add that he was willing to discuss language to clarify the deal. Supporters of the package, both in union leadership and in the administration, have said they believe one factor behind its rejection was rank-and-file misunderstanding over proposed health care changes.

The proposed concessions also included included a two-year wage freeze, new restrictions on pension and other retirement benefits and various management-labor cost-saving committees

“I certainly have to understand before we clarify or correct what the road map to an approval process would look like,” the governor added. “Having said that, you never close the door.”

Spokesmen for the State Employees Bargaining Agent Coalition confirmed Tuesday that union leadership has begun a review of labor bylaws to consider asking members to reform the process for considering contract amendments. But SEBAC spokesman Matt O’Connor and Larry Dorman declined to provide any specific details.

Malloy did not suggest how SEBAC’s voting rules should be changed, but he said the current threshold for adopting contract amendments is unworkable. Eleven of 15 member unions endorsed the tentative concession deal, with 57 percent of members casting ballots voting in favor, but the threshold for approval is at least 14 unions representing 80 percent of the membership.

The governor also said that unions don’t have to wait until the bylaw review process is complete and reforms potentially are adopted before sharing their game plan with him. “It doesn’t have to be written in stone,” he said. “But we can’t go down that road without understanding what the approval process would be like.”

O’Connor said Wednesday that “we understand the governor’s interest in the approval process — and his recognition that any change to the process is an internal SEBAC matter.”

O’Connor added that “our preference, however, is to begin discussions with the administration in the meantime. Some of the governor’s reported press comments suggest a willingness to do that. Given how much is at stake for the economy, for the critical public structures upon which the economy depends, and for our members and the services they provide, we think time is working against all of us.”

The administration must submit a plan to the General Assembly by July 15 to save $700.7 million this fiscal year and $901.2 million in 2012-13.

Malloy’s budget staff recommended about 6,675 layoffs and elimination of roughly 1,000 vacant positions to help meet those targets. Though departments have until Friday to submit alternative plans, those are expected only to modestly shrink or increase the number of planned jobs cuts, not eliminate the need for them altogether.

“It’s not as if I’m not desirous of finding a solution,” the governor said. “This is not a time to be laying off 6,500 people.”