The loan move is part of an agreement between La Liga, the General Sports Authority of Saudi Arabia and the Federation of Saudi Arabia to give Saudi players European experience ahead of the World Cup in Russia this summer.
The 26-year-old winger will join from Al-Hilal, where he has played a total of 161 games in which he scored 28 goals and has made 23 assists.
Al-Dawsari has made 17 appearances for his country, scoring two goals.
He is one of several players to make a move to the Spanish league this weekend, with Yahia Al-Shehri also moving on loan to Leganés from Al-Nasr and Nouh Al-Mousa, who has gone on loan at Real Valladolid from Al-Fateh.
Levante from the top flight, alongside second division outfits Gijon and Rayo Vallecano also simultaneously announced on Sunday they had signed a Saudi Arabian player.
For the Spanish clubs, the loans are a way to gain exposure in a little-tapped market.
"With Yahia Al-Shehri's arrival, the club has improved its squad and also gained visibility in the Arab world," Leganes said in a statement.