In the context of the debate on cryptocurrencies as the ‘digital gold’, this study explores the nexus between the Bitcoin and US oil returns by employing a rich set of parametric and non-parametric approaches. We examine the dependence structure of the US oil market and Bitcoin through Clayton copulas, normal copulas, and Gumbel copulas. Copulas help us to test the volatility of these dependence structures through left-tailed, right-tailed or normal distributions. We collected daily data from 5 February 2014 to 24 January 2019 on Bitcoin prices and oil prices. The data on bitcoin prices were extracted from coinmarketcap.com. The US oil prices were collected from the Federal Reserve Economic Data source. Maximum pseudo-likelihood estimation was applied to the dataset and showed that the US oil returns and Bitcoin are highly vulnerable to tail risks. The multiplier bootstrap-based goodness-of-fit test as well as Kendal plots also suggest left-tail dependence, and this adds to the robustness of the results. The stationary bootstrap test for the partial cross-quantilogram indicates which quantile in the left tail has a statistically significant relationship between Bitcoin and US oil returns. The study has crucial implications in terms of portfolio diversification using cryptocurrencies and oil-based hedging instruments.