FDA has approved a saliva based test that was funded by the NBA and NBAPA.
The test, known as SalivaDirect, is designed for widespread public screening. The cost per sample could be as low as about $4, though the cost to consumers will likely be higher than that -- perhaps around $15 or $20 in some cases, according to expert sources.
https://www.espn.com/nba/story/_/id/29667299/fda-allowing-saliva-based-test-funded-nba
[Edit: some more info on this
General article on saliva testing https://www.the-scientist.com/news-opinion/saliva-tests-how-they-work-and-what-they-bring-to-covid-19-67720
Test U of Illinois plans to use https://www.biorxiv.org/content/10.1101/2020.06.18.159434v1.full.pdf
Yale/NBA developed test https://www.medrxiv.org/content/10.1101/2020.08.03.20167791v1
Note that the Yale and the U of Illinois test skip a step in the standard Covid-19 test - RNA extraction - but are still RT-PCR based tests]