Vision Transformers (ViTs) have been widely used in large-scale Vision and Language Pre-training (VLP) models. Though previous VLP works have proved the effectiveness of ViTs, they still suffer from computational efficiency brought by the long visual sequence. To tackle this problem, in this paper, we propose an efficient vision-and-language pre-training model with Text-Relevant Image Patch Selection, namely TRIPS, which reduces the visual sequence progressively with a text-guided patch-selection layer in the visual backbone for efficient training and inference. The patch-selection layer can dynamically compute text-dependent visual attention to identify the attentive image tokens with text guidance and fuse inattentive ones in an end-to-end manner. Meanwhile, TRIPS does not introduce extra parameters to ViTs. Experimental results on a variety of popular benchmark datasets demonstrate that TRIPS gain a speedup of 40% over previous similar VLP models, yet with competitive or better downstream task performance.