Let \((J_n)\) be the sequence of Bessel functions of integer order, and let \((O_n)\) be the sequence of Neumann polynomials, i.e. \[ O_0 (z)= {1\over z}, \qquad O_n (z)= {1\over 4} \biggl( {2\over z} \biggr)^{n+1} \sum_{\nu =0}^{[ n/2 ]} {{n(n- \nu- 1)!} \over {\nu!}} \biggl( {z\over 2} \biggr)^{2\nu} \] for \(n\in \mathbb{N}\). The author considers series expansions of the form \[ \sum^\infty_{n=0} (A_n (f) J_n (z)+ B_n (f) O_n (z)), \qquad |z|=1, \tag{1} \] where \(f\) is Lebesgue integrable on \(|z|=1\), called Neumann-Bessel series. He obtains an (approximate) expression for the kernel functions of the partial sums of (1), and he describes some equiconvergence between the partial sums of (1) and the partial sums of the Fourier series of \(f\). Finally, he gives various applications, where he also considers the case of usual Neumann series \(\sum^\infty_{n=0} A_n (f) J_n (z)\).