Carbon dioxide (CO₂) and methane (CH₄) are the most important anthropogenic greenhouse gases and the main drivers of climate change. Monitoring their concentrations from space helps to detect and quantify anthropogenic emissions, supporting the mitigation efforts urgently needed to meet the primary objective of the United Nations Framework Convention on Climate Change (UNFCCC) Paris Agreement to limit the global average temperature increase to well below 2 °C above pre-industrial levels. In addition, satellite observations can be used to quantify natural sources and sinks improving our understanding of the carbon cycle. Advancing these goals is the motivation for the European Copernicus CO₂ monitoring mission CO2M. The necessary accuracy and precision requirements for the measured quantities XCO2 and XCH4 (the column-averaged dry-air mixing ratios of CO₂ and CH₄) are demanding. According to the CO2M mission requirements, the spatial and temporal variability of the systematic errors of XCO2 and XCH4 shall not exceed 0.5 ppm and 5 ppb, respectively. The stochastic errors due to instrument noise shall not exceed 0.7 ppm for XCO2 and 10 ppb for XCH4. Conventional so-called full-physics algorithms for retrieving XCO2 and/or XCH4 from satellite-based measurements of reflected solar radiation are typically computationally intensive and still usually require empirical bias corrections based on supervised machine learning methods. Here we present the retrieval algorithm NRG-CO2M (Neural networks for Remote sensing of Greenhouse gases from CO2M), which derives XCO2 and XCH4 from CO2M radiance measurements with minimal computational effort using artificial neural networks (ANNs). Since CO2M will not be launched until 2026, our study is based on simulated measurements over land surfaces from a comprehensive observing system simulation experiment (OSSE). We employ a hybrid learning approach that combines advantages of simulation-based and measurement-based training data to ensure coverage of a wide range of XCO2 and XCH4 values making the training data also representative of future concentrations. The algorithm's postprocessing is designed to achieve a high data yield of about 80 % of all cloud-free soundings. The spatio-temporal systematic errors of XCO2 and XCH4 amount 0.44 ppm and 2.45 ppb, respectively. The average single sounding precision is 0.41 ppm for XCO2 and 2.74 ppb for XCH4. Therefore, the presented retrieval method has the potential to meet the demanding CO2M mission requirements for XCO2 and XCH4.