Kentucky, by contrast, announced on Monday that it was delaying its primary from May 19 to June 23. The announcement was made by Kentucky’s secretary of state, Michael G. Adams, in a video statement on Twitter.

“Postponing the primary was not an easy decision, but the Republican secretary of state and Democratic governor agreed, and so do county clerks from both parties,” Mr. Adams said. “My hope is that this delay will allow us to have a normal election. Even if not, this delay will allow me, the state board of elections and our county clerks time to assess what changes we must make to ensure a successful primary election.”

The proposed new date for the Kentucky primary, June 23, would fall outside the June 9 deadline set by the Democratic National Committee to hold all nominating contests, and could result in a loss of delegates.

The moves to postpone elections on Monday came days after elections officials in Louisiana and then Georgia said they would push back their upcoming primary elections in response to the outbreak. On Friday, Louisiana announced that its April 4 election would be postponed by more than two months, to June 20; Georgia followed suit a day later, saying it would reschedule its March 24 contest for May 19.

Kate Bedingfield, deputy campaign manager for former Vice President Joseph R. Biden Jr., said in a statement after Mr. DeWine’s afternoon announcement, “We will follow the guidance offered by state public health officials for how to best ensure their populations are looked after while encouraging participation in our democracy.” Senator Bernie Sanders’s campaign declined to comment.

The developments in Ohio added yet another logistical speed bump to a presidential primary that has already been upended by concerns that voting could worsen a public health crisis.

Several other states have said they were changing or considering making changes to their voting processes and procedures in response to the outbreak. Wyoming suspended the in-person portion of its Democratic caucuses, allowing residents to either vote by mail or drop off their ballots at a county polling location instead.