Remote sensing can be used for oil spill detection. To minimize the impact of oil pollution on the ecosystems, it is imperative that oil spills are detected at the earliest possible stage in order that the relevant monitoring frameworks can be put in place and appropriate response measures initiated. This paper presents two different approaches for oil spill detection on optical satellite imagery from the Landsat-8 and Landsat-9 satellites using deep learning techniques. This comprises the application of a (fully connected) deep neural network (DNN) and a convolutional neural network (CNN) in the type of a U-Net architecture. The models were developed to recognise and classify patterns of oil spills against the complex background of marine and coastal environment. Consequently, the performance of the models is evaluated and their efficiency demonstrated on different datasets. The experimental results indicate usability of the analysed methods. This study is based on a limited amount of manually labelled training data and serves to validate the potential of deep learning based oil spill detection on optical satellite remote sensing images.