Rep. Carolyn Cheeks Kilpatrick said the primary would be statewide and would be funded through party money. She is one of four Democrats who are not endorsing either candidate who are working together on a plan for a repeat primary.
Amid talks with both campaigns, the four Michigan Democrats said in a statement they were "focusing on the possibility of a state-run primary in early June which would not use any state funding."
"This option would require the passage of legislation by the State Legislature, and we look forward to working with the members of the Legislature in the coming days to see if this option can be made a reality," the Democrats said.
Clinton won a primary the state held in January, but Obama's name wasn't on the ballot. He had it removed because the primary was held too early to comply with national party rules, meaning no delegates were at stake.
Florida also had its delegates stripped for holding a primary in January. The Florida Democratic Party offered a plan for a mail-in primary Thursday, but admitted it didn't have a good chance of being approved. Kilpatrick said the mail-in approach wouldn't work in Michigan.
Kilpatrick said discussions on the plan would continue Friday. "We're hopeful. We're close," she told reporters after appearing at a Judiciary Committee hearing on Capitol Hill.
Other Michigan Democrats working on the plan include Democratic National Committee member Debbie Dingell, Sen. Carl Levin and United Auto Workers President Ron Gettelfinger.
Kilpatrick said although she is optimistic, Detroit City Clerk Janice Winfrey has expressed concern they may not have the time or manpower to pull it off. She said they are trying to work through those issues.
To go forward, any plan would require the approval of the two candidates' campaigns, the Democratic National Committee, state party leaders and Gov. Jennifer Granholm, who is backing Clinton.
Michigan Democrats need to act quickly because the politically divided legislature will have to sign off on the deal and approve how to spend the privately raised funds for a new election. Members of the Democratic-controlled state House and Republican-controlled state Senate leave at the end of the month on their two-week spring break.
Florida Democrats said they will make a decision by Monday on whether to hold a dual mail-in and in-person re-vote. But the plan floated this week faces opposition from the state's Democratic congressional delegation, and Obama has also expressed concerns about security and accuracy of a mail-in vote organized so quickly. Democratic National Committee rules require the vote to be scheduled by June 10.