He earlier issued an order against such policies in schools, but some districts have pushed back, saying their pupils must still mask.
The row comes as US schools begin their new academic year, and as Delta variant surges affect younger groups.
There is no evidence the Delta variant is more severe in children.
Multiple states have recently recorded spiking paediatric hospital admissions, though a number of the child patients reportedly have other conditions such as asthma, diabetes or respiratory syncytial virus.
Just 1% of US children infected with coronavirus end up in hospital and 0.01% die, according to the American Academy of Pediatrics.
Across the US, there have been showdowns between governors and local school districts over student mask mandates.
In a statement on Monday, the Republican governor of Florida said the state board of education “could move to withhold the salary of the district superintendent or school board members” if they issue rules requiring children to wear face masks. Teachers or other school staff would not be affected.
In Mr DeSantis’ July order, he said forcing masks on students would encroach on parents’ rights.
The same day, one of Florida’s largest government-run school districts announced that students must wear masks for the beginning of the school year, unless they have a excuse from a doctor or psychologist.
“Heaven forbid we lost a child to this virus, I can’t just simply blame the governor of the state of Florida – I just can’t,” said Leon County Superintendent Rocky Hanna during a live-streamed news conference to announce the decision.
Another Florida superintendent cautioned in a Monday opinion piece for the Washington Post that two employees have died of the virus in the last two weeks, “and school hasn’t even started”.
Board members in Broward County, the state’s second-largest school district, are to discuss the issue on Tuesday.
Public health experts and teachers say masks are needed to prevent the spread of the highly contagious Delta variant – especially as the vaccines are not yet approved for children under the age of 12.
The US Centers for Disease Control (CDC) says students above the age of two and all staff should mask indoors even if they are vaccinated, and socially distance in classrooms to reduce risk.
More than 13,600 people in Florida were in hospital with Covid-19 as of Monday, according to federal health authorities. There are over 100 children currently in hospital due to the virus, local media report.
About half of the Sunshine State’s population has been fully vaccinated.
It is one of several states in the southern US to see dramatic rises in virus cases and hospital admissions in recent weeks.
On Monday, school districts in Dallas and Austin – two of the largest in Texas – announced they were imposing mask requirements in defiance of Governor Greg Abbott, a Republican.
In response, Governor Abbott’s office said the time for mask mandates was “over”. The statement added that Mr Abbott believes that parents and guardians should have the right to decide whether a child wears a mask, “as with any other decision in their child’s life”.
On Monday, Arkansas’ Republican governor warned only eight intensive care hospital beds remained in the entire state amid a record-breaking surge in Covid-related admissions.
Louisiana, which is governed by a Democrat, also has low vaccination rates. It has reported 2,700 Covid patients in hospital as of last week – a return to 2020-levels.