Recently, some works integrate SSL techniques into deep clustering frameworks to enhance image clustering performance. However, they all need pretraining, clustering learning, or a trained clustering model as prerequisites, limiting the flexible and out-of-box application of SSL learners in the image clustering task. This work introduces ASD, an adaptor that enables the cold-start of SSL learners for deep image clustering without any prerequisites. Specifically, we first randomly sample pseudo-labeled data from all unlabeled data, and set an instance-level classifier to learn them with semantically aligned instance-level labels. With the ability of instance-level classification, we track the class transitions of predictions on unlabeled data to extract high-level similarities of instance-level classes, which can be utilized to assign cluster-level labels to pseudo-labeled data. Finally, we use the pseudo-labeled data with assigned cluster-level labels to trigger a general SSL learner trained on the unlabeled data for image clustering. We show the superior performance of ASD across various benchmarks against the latest deep image clustering approaches and very slight accuracy gaps compared to SSL methods using ground-truth, e.g., only 1.33% on CIFAR-10. Moreover, ASD can also further boost the performance of existing SSL-embedded deep image clustering methods.