Hi micbar, one of the reputable ones is Add International. They deployed me several years ago for Hospital in Saudi. That was years ago so I'm not really sure if they are still hiring for Saudi nowadays. However, it is worth trying.
If you're staying in manila. Just take a walk in Malate.. you'll find several agencies there who are recruiting nurses for saudi arabia. Also try Abba International which is located in Ortigas Goldloop Tower.