British media reports said Newcastle will pay an initial fee of 30 million pounds, with additional performance-related payments included in the deal, for the 20-year-old.
Bamba is a graduate of Monaco's academy and broke into the club's first team last season. He made 25 appearances in all competitions, including both legs of Monaco's UEFA Champions League knockout phase play-off against Paris Saint-Germain.
The young midfielder said he is looking forward to testing himself in England after his breakthrough campaign in France.
"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.
Newcastle manager Eddie Howe believes the midfielder has the qualities to strengthen his squad while continuing to develop his game.
"Aladji adds depth and a range of qualities to our midfield options, with huge potential for growth," Howe said.
"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."
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