(CNN) -- A proposal is taking shape for Michigan Democrats to hold a new presidential primary, a Democratic source close to the negotiations says.
Voters go to the polls in Warren, Michigan, in the January 15 presidential primary.
The national Democratic Party has refused to recognize the results of the first vote, held two months ago.
Under the proposal, Michigan would hold another primary on June 3.
The Democratic Party would reimburse the state for the cost of running the election, ensuring that cash was in hand before the voting began.
The Democratic National Committee did not recognize the results of the January 15 Michigan primary. The DNC had ruled that only four states could hold their contests before February 5, but Michigan wasn't one of those.
Florida is in much the same situation as Michigan, holding its primary on January 29 in defiance of DNC rules.
After the DNC decision on Michigan, Obama was among the Democratic presidential contenders who had their names taken off the state's ballot. However, Clinton didn't remove her name and won 55 percent of the vote. Some 40 percent of Democrats in the state filed ballots declaring themselves "uncommitted."
Michigan voters wanting to cast ballots in the new primary would have to identify themselves as Democrats and certify that they did not vote in the state's Republican primary in January, the source said. Michigan usually does not require party identification in primary elections.
Exit polling indicated 79 percent of voters in the January Democratic primary were Democrats, 3 percent were Republicans and 18 percent were independents.
"Not all the campaigns are happy" with the proposal at this point, the source close to the negotiations said.
Rep. Carolyn Cheeks Kilpatrick, D-Michigan, said she is working with state Democratic leaders and the Clinton and Obama campaigns on a plan for another vote to ensure that all of Michigan's 156 delegates get seated at the national convention in August.
"We are trying to get there. It's not a done deal yet," Kilpatrick said.
Kilpatrick described three other options on the table: One includes a "redo," with the state paying the cost. The second is a mail-in primary, which the lawmaker said was "too hard to do." The third was to "have the two candidates get together" and agree to a way to apportion the delegates.
Kilpatrick said more discussions are planned Friday afternoon on the June 3 proposal, but she added, "The DNC will make the final decision."
Kilpatrick, chairwoman of the Congressional Black Caucus, has not endorsed a candidate.
In Florida, the party this week floated a plan that called for a June 3 election with mail-in ballots as well as precinct voting.
But the state's Democratic congressional delegation rejected that proposal on Thursday.
House Democrats said they were opposed to any revote -- particularly one involving mailed ballots.
"After reviewing the party's proposal and individually discussing this idea with state and local leaders and elections experts, we do not believe that this is a realistic option at this time and remain opposed to a mail-in ballot election or any new primary election in Florida of any kind," said a statement from the delegation. E-mail to a friend
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