•We investigate co-movement between oil and stock market in time–frequency space.•Crude oil is the main driver of co-movement in the mid- and long-term scales.•Crude oil rarely affects the co-movement in oil-importing countries’ stock markets.•GCC stock markets lead over oil-importing countries’ stock markets in the long term. In this paper, we employ partial- and multiple-wavelet coherence analyses to examine co-movement between international stock markets by considering the influence of crude oil in a time domain perspective. Overall, we find that crude oil is a major factor driving co-movement between international stock markets in the median and long term. However, when considering the oil-importing and oil-exporting countries differently, we still find that crude oil is a driver for interdependence between oil-importing and oil-exporting countries. In contrast, the crude oil has relative lower impact on the co-movement in oil-importing or in oil-exporting countries, which indicates its co-movement is caused by other factors. In addition, Gulf Cooperation Council stock market may lead the stock markets of oil-importing countries in the long term. Our empirical results provide meaningful information for investors and policymakers.