WASHINGTON—The drive for a second Michigan presidential primary collapsed Thursday, and a fresh dispute broke out between Barack Obama and Hillary Rodham Clinton over the fate of the state’s 128 national convention delegates.

Obama’s campaign said a fair resolution would be to split them evenly with Clinton. Aides to the former first lady instantly rejected the idea and said they would consider a mail-in primary—even though Obama has raised concerns about the security of a vote by mail organized so quickly.

Obama leads the overall competition for convention delegates, and Clinton has been leading the effort to hold a revote in Michigan, eager for a chance to close the gap on her rival.

The state held a primary in January, so early in the year that it violated party rules. As a result, it was stripped of its delegates. Obama and several other Democratic candidates removed their name from the Michigan ballot and all Democratic candidates agreed not to campaign there.

Michigan Democratic leaders had proposed a do-over primary, to be conducted by the state on June 3 with funding by private donors. But lawmakers showed a lack of enthusiasm for a bill authorizing the vote, and state Senate adjourned Thursday without taking it up.