Less than eight weeks before Floridians will vote on an abortion-rights amendment to the state Constitution, a bitter standoff is escalating between Gov. Ron DeSantis and the amendment’s backers over whether a string of state actions directed at the measure amount to a taxpayer-funded effort to defeat it at the polls.

The debate first erupted last week after the state Agency for Health Care Administration posted a 30-second video on social media that casts current Florida law — which bans almost all abortions after six weeks of pregnancy — as proof that “Florida cares about women and families.” The post links to an agency website that claims that the abortion rights measure, known as Amendment 4, “threatens women’s safety.” It adds: “Don’t let the fear-mongers lie to you.”

At the same time, state criminal investigators were revealed to be belatedly examining whether some of nearly 911,000 signatures collected to place the measure on the November ballot might be fraudulent. State officials certified the signatures in February, and the deadline for challenging them has passed.

This week, the state health care agency has begun airing its social-media video as a commercial on Florida television, although the extent and cost of the ad campaign remains unclear. The agency’s media office did not address questions about the campaign, referring instead to a statement by a DeSantis official about the value of public-service announcements.

On Tuesday, the American Civil Liberties Union of Florida, said it was preparing to file a lawsuit in state court on behalf of Floridians Protecting Freedom, which is campaigning for the amendment. . The A.C.L.U. said the suit would allege that the agency is illegally spending taxpayer dollars on a political campaign against the measure.

“The government is trying every dirty trick they can to distract Floridians,” Lauren Brenzel, the campaign director at Floridians Protecting Freedom, said on Tuesday at a news conference. “They’re doing everything they can to suppress the votes and the will of the people.”