We determine the optical properties of whole blood samples in the near infrared spectral range from double integrating sphere measurements using an inverse Monte Carlo technique. The measured values included the diffuse reflectance, the total transmittance, and the collimated transmittance. From these data, the absorption coefficient, the scattering coefficient, and the anisotropy factor were derived. The spectral range investigated extended from 700 nm to 1200 nm. It was found that the optical properties of blood were substantially different from the respective data for other relevant human tissues known so far. In addition, we analyzed the effect of the scattering phase function approximation on the resulting estimates of the optical properties. The Henyev-Greenstein and the Gegenbauer kernel phase functions were considered. The calculated angular distributions of scattered light were compared with goniophotometric measurements performed at the wavelength of 633 nm. The data presented in this study prove that the variations of the employed scattering phase function approximation can cause large discrepancies in the derived optical properties. This leads to the conclusion that the exact knowledge of the scattering phase function is required for the precise determination of the optical constants from the double integrating sphere measurements.