The Amplified Source Multiplication (ASM) method and its improved Modified Source Multiplication (MSM) method have been widely used in the CEA's EOLE and MASURCA critical facilities over the past decades for the determination of reactivity worths by using fission chambers in subcritical configurations. The ASM methodology uses relatively simple relationships between count rates of efficient miniature fission chambers located in slightly subcritical reference and perturbed configurations. While this method works quite well for small reactivity variations, the raw results need to be corrected to take into account the flux perturbation at the fission chamber location. This is performed by applying to the measurement a correction factor called MSM. This paper describes in detail both methodologies, with their associated uncertainties. Applications on absorber cluster worth in the MISTRAL-4 full MOX mock-up core and the last core loaded in MASURCA show the importance of the MSM correction on raw ASM data.