The club will pay an initial fee of £30 million, plus add-ons, for the 20-year-old, according to British media reports.
A product of Monaco's academy, Bamba made his first-team debut for the Ligue 1 club last year and went on to make 25 appearances, including both legs of Monaco's Champions League knockout phase play-off against Paris St Germain.
“I’m excited to test myself in a new league, the biggest league in the world. I learned a lot last season and now I feel ready for this new opportunity. I will give my all for this club,” Bamba said in a statement.
Newcastle boss Eddie Howe added: “Aladji adds depth and a range of qualities to our midfield options, with huge potential for growth. His development since breaking through at Monaco has been really impressive and something we believe we can help him build on.
“He operates in a deeper role but has the ability to contribute all over the pitch, which he demonstrated in both Ligue 1 and the Champions League last season."
As the summer transfer window ramps up, Flashscore has everything you need to keep up with breaking news, rumours and analysis. Follow our live transfer blog to follow the biggest moves, whilst expert Dean Jones will be on hand to provide the inside stories from around the world.