A main drawback of near-field measurement techniques is the long measurement time which is a strong limiting factor in many measurements. The reason for this is the large number of required measurement points and a reduction without loss of accuracy is hence desirable. In many spherical near-field measurements the relevant information is mostly concentrated in a few spherical wave coefficients. Methods from compressed sensing (CS) exploit this structure to reduce measurement efforts if it is properly adapted to the theory of spherical near-field to far-field transformation. In this paper, requirements and challenges of CS application in this problem will be summarized and discussed. Simulation results indicate the potential capabilities of CS although further research is needed to provide the theoretical guarantees and to investigate limitations and uncertainties.