St. Paul, Minnesota (CNN) -- Minnesota Gov. Mark Dayton said he thinks he and legislators are out of "good" options in the battle to bring Minnesota's government out of a shutdown that's been in place since Friday morning, though he remains "committed."
Dayton, a Democrat, and the leaders of the Republican-controlled legislature, Sen. Amy Koch and Rep. Kurt Zellers, met for just over an hour Tuesday -- the first time the three had met since their eleventh-hour attempts to avoid a shutdown Thursday night.
"I'm not hopeful any more than I was before (this) meeting," Dayton told reporters following the closed-door meeting, "because I think we've got the same gulf between us that we've had all along."
He was referring to the state's $5 billion budget gap and the disagreement Democrats and Republicans have had over how to fix it. Dayton had asked for a plan that includes increasing taxes on the state's millionaires. Republicans say they won't agree to anything involving new taxes.
The Minnesota government virtually shut down on Friday, leaving only a limited array of state services in operation over the busy holiday weekend.
Many social service agencies will lose their funding, cutting state support for programs such as job training and homelessness prevention. Those that don't have reserves will likely close their doors. Road construction projects will cease, as will licensing for teachers and businesses.
Up to 23,000 state workers are scheduled to be laid off, though they will continue to get health benefits and can return to their jobs when the budget impasse is resolved.
Basic health and safety services will continue, a judge ruled on Wednesday.
One area that has caused significant stress between both sides is the Health and Human Services bill.
"That (bill) is where all the growth is, that's where the most reform is needed," Republican Senate Majority Leader Koch said, adding that they believe certain bills "could have been easily wrapped up" as early as last week, which would have allowed government to continue functioning.
Republicans have called this plan the "lights on" bill that essentially would have kept government up and running while the troublesome bills and budget areas remained in discussion.
Dayton was not sold on that idea last week, and he remained opposed to it Tuesday, citing a lack of enough common ground on both policy and budgetary issues in any of the bills.
Talks will continue. Their next meeting is planned for Wednesday afternoon. Dayton said meetings with legislators involved with Health and Human Services and possibly education would convene Thursday morning to discuss those more troublesome areas.
"We're all committed to continuing this process," Dayton added. "There's too much at stake for the people of Minnesota not to give it every possible effort continuously until it's resolved."
CNN's Tami Luhby contributed to this report.