TALLAHASSEE — Florida’s House and Senate leaders are throwing cold water on headline-grabbing bills that would eliminate no-excuse mail voting and impose a near-total abortion ban with no exceptions for rape and incest.

The proposals are unlikely to succeed during the 60-day session that started Tuesday without their support.

House Speaker Paul Renner and Senate President Kathleen Passidomo said they don’t want to significantly scale back mail-in voting or further restrict abortion.

“We have a lot of elderly people who like the vote-by-mail process,” said Passidomo, R-Naples.

A bill (SB 1752) by state Sen. Blaise Ingoglia, R-Spring Hill, seeks to re-establish old absentee ballot rules. To vote by mail, Floridians would have to sign a certificate swearing they have a reason for voting absentee, such as an illness or disability, and face being charged with a felony if they lied about it.

Ingoglia, a former chairman of the Florida GOP, said that he thought it was “time to go back to basics” and casting an in-person ballot was the safest way of voting.

But Renner, R-Palm Coast, said he didn’t see a need to end no-excuse mail-in voting.

“I am not looking to recede from the easier-to-vote piece,” he said. “I would rather look at let’s just make sure we have got the controls in place to make sure it is still harder to cheat.”