It has already been approved by the Department of Education, which like the board is led by DeSantis appointees, and will not need legislative approval to take effect, according to the reports.
If approved, teachers will be prohibited from "intentionally" teaching topics related to "sexual orientation or gender identity" to students from the fourth grade through to their final year of high school.
Last year's initiative applied only to kindergarten through third grade.
"Make no mistake: This is a part of a disturbing and dangerous trend that we're seeing across the country of legislations that are anti-LGBTQI+, anti-trans," said White House Press Secretary Karine Jean-Pierre on Wednesday.
DeSantis, who won reelection in 2022 in a landslide victory, has made battling politicians, teachers and businesses he accuses of wanting to impose a progressive "woke" ideology on others a central concern of his second term.
He has increasingly courted conservative voters with controversial proposals on education and immigration in recent months, as jockeying for the 2024 Republican presidential nomination heats up.
A rising star of the American right, DeSantis is widely expected to go toe-to-toe with former president Donald Trump, who has already declared his candidacy.
The move to expand the "Don't Say Gay" restrictions has already drawn pushback from activists, including Equality Florida.
"This has been the goal all along: sweeping censorship and book banning targeting LGBTQ people in service to his presidential ambitions," the group tweeted Wednesday, referring to DeSantis.
"Now educators, in any grade level, and their livelihoods are being placed directly in the crosshairs for acknowledging that LGBTQ people exist."