Summary: We consider partitioned lists of real numbers \(\Lambda = \{\lambda_1,\lambda_2,\dots,\lambda_n\}\), and give efficient and constructive sufficient conditions for the existence of nonnegative matrices with spectrum \(\Lambda\). Our results extend the ones given in [the first two authors, Linear Algebra Appl. 416, No. 2–3, 844–856 (2006; Zbl 1097.15014); the first two authors, J. Moro and A. Borobia, Electron. J. Linear Algebra 16, 1–18 (2007; Zbl 1155.15010)] for the real and symmetric nonnegative inverse eigenvalue problem. We also consider the complex case and show how to construct an nonnegative \(r\times r\) matrices with prescribed complex eigenvalues and diagonal entries.