WASHINGTON — Even the scoreboards in high school gyms eventually will have to promote good health.

Moving beyond the lunch line, new rules that will be proposed Tuesday by the White House and the Agriculture Department would limit marketing of unhealthy foods in schools. They would phase out the advertising of sugary drinks and junk foods around campuses during the school day and ensure that other promotions in schools were in line with health standards that already apply to school foods.

That means a scoreboard at a high-school football or basketball game eventually wouldn’t be allowed to advertise Coca-Cola, for example, but it could advertise Diet Coke or Dasani water, which is also owned by Coca-Cola. Same with the front of a vending machine. Cups, posters and menu boards advertising foods that don’t meet the standards also would be phased out.

Ninety percent of such marketing in schools is related to beverages, and many soda companies already have started to transition their sales and advertising in schools from sugary sodas and sports drinks to their own healthier products.

The proposed rules are part of first lady Michelle Obama’s Let’s Move initiative to combat child obesity, which is celebrating its fourth anniversary this week.

The rules also would allow more children access to free lunches and ensure that schools have wellness policies in place.

The proposed rules come on the heels of USDA regulations that are now requiring foods in the school lunch line to be healthier.

Rules set to go into effect next school year will make other foods around school healthier as well, including in vending machines and separate “a la carte” lines in the lunch room. Calorie, fat, sugar and sodium limits will have to be met on almost every food and beverage sold during the school day. Concessions sold at afterschool sports games would be exempt.

The healthier food rules have come under fire from conservatives who think the government shouldn’t dictate what children eat — and from some students who don’t like the healthier foods.

Aware of the backlash, the USDA is allowing schools to make some of their own decisions on what constitutes marketing and asking for comments on some options.

Rules for other school fundraisers, like bake sales and marketing for those events, would be left up to schools or states.