A number of studies have successfully developed speaker verification or spoofing detection systems. However, studies integrating the two tasks remain in the preliminary stages. In this paper, we propose two approaches for the integrated replay spoofing-aware speaker verification task: an end-to-end monolithic and a back-end modular approach. The first approach simultaneously trains speaker identification, replay spoofing detection, and the integrated system using multi-task learning with a common feature. However, through experiments, we hypothesize that the information required for performing speaker verification and replay spoofing detection might differ because speaker verification systems try to remove device-specific information from speaker embeddings while replay spoofing exploits such information. Therefore, we propose a back-end approach using a deep neural network that takes speaker embeddings extracted from enrollment and test utterances and a replay detection prediction on the test utterance as input. Experiments are conducted using the ASVspoof 2017-v2 dataset, which includes official trials on the integration of speaker verification and replay spoofing detection. The proposed back-end approach demonstrates a relative improvement of 21.77% in terms of the equal error rate for integrated trials compared to a conventional speaker verification system.