The Magpies agreed the deal in principle with Belgian club Anderlecht last weekend, but Mbemba was unable to seal his move while he waited for paperwork issues to be resolved.
The 20-year-old Congo DR international has now been granted governing body endorsement (GBE) at a Football Association panel hearing on Thursday, and will be able to launch his Newcastle career once he gets visa clearance.
"We are delighted that Chancel is now a Newcastle United player after it took a little longer than we had hoped," McClaren told Newcastle's website.
"He is a proven performer at the highest level, he's powerful, combative and will add quality to our defence. We will get him integrated into the squad quickly once he is with us.
"Unfortunately, due to the time it takes to process the visa application that won't be in time for our final friendly on Saturday, but our focus will be to make sure he will be ready and available as soon as possible."