/ AbstractUp to 150000 asteroids will be visible in the images of the ESA Euclid space telescope, and the instruments of Euclid offer multiband visual to near-infrared photometry and slitless spectra of these objects. Most asteroids will appear as streaks in the images. Due to the large number of images and asteroids, automated detection methods are needed. A non-machine-learning approach based on the StreakDet software was previously tested, but the results were not optimal for short and/or faint streaks. We set out to improve the capability to detect asteroid streaks in Euclid images by using deep learning. We built, trained, and tested a three-step machine-learning pipeline with simulated Euclid images. First, a convolutional neural network (CNN) detected streaks and their coordinates in full images, aiming to maximize the completeness (recall) of detections. Then, a recurrent neural network (RNN) merged snippets of long streaks detected in several parts by the CNN. Lastly, gradient-boosted trees (XGBoost) linked detected streaks between different Euclid exposures to reduce the number of false positives and improve the purity (precision) of the sample. The deep-learning pipeline surpasses the completeness and reaches a similar level of purity of a non-machine-learning pipeline based on the StreakDet software. Additionally, the deep-learning pipeline can detect asteroids 0.25-0.5 magnitudes fainter than StreakDet. The deep-learning pipeline could result in a 50% increase in the number of detected asteroids compared to the StreakDet software. There is still scope for further refinement, particularly in improving the accuracy of streak coordinates and enhancing the completeness of the final stage of the pipeline, which involves linking detections across multiple exposures.