Wordpress Login using member info from an external database (web application)

Hi! I am web designer and developer and my team is currently working on a Wordpress website that will have private pages. This private pages will only be accesible to certain members... but I will not be using the wordpress member / user system. Currently our client has a php website where certain users (property owners) login and they access some information like user information like their personal information and contract. But this area is not going to be used anymore. Is old and the only thing that will be running will be the database. This new site I am building will be the new site where owners will be able to login and acces the information. That means this new site will need to access to the other old database just to authenticate login credentials and then come back to the wordpress website so the logged in owner can see private info form private pages. Remember this website will be a public website, so anyone can see it. But it will also have private pages that only members registered in that other database can see. I have direct contact with the support persons from that old database site so they can give me anything or any access I want. The persons of that system recommended us using web apps for authentication. But I have no experience in that. I will give you the wordpress installation and whateveryou need from the other server to authenticate users. If you think wordpress is not the best platform and instead use custom php files, that can be considered too we are just in the phase of design. I will add this as a simple project because of my lack of experience but if you suggest the budget is other I will gladly change it. Thank you!