We extend a recent approach to observer design for linear differential-algebraic systems to systems with Lipschitz nonlinearities. The observer design further extends the standard Luenberger-type observer design. We show that the design parameters for the observer can be obtained by the solution of a Riccati-type inequality. The solutions of the latter can in turn be obtained by solving a set of linear matrix inequalities (LMIs) and bilinear matrix inequalities (BMIs) which provides a computational procedure. A feature of the presented observer design is the possibility of reformulation as an ordinary differential equation.