We use the convolution method for arithmetic functions of several variables to deduce an asymptotic formula for the number of k-tuples of positive integers with components which are pairwise non-coprime and ≤ x. More generally, we obtain asymptotic formulas on the number of k-tuples (n₁, ... , n_k) ∈ N^k such that at least (respectively, exactly) r pairs (n_i, n_j) are coprime. Our results answer the questions raised by Moree (2005, 2014), and generalize and refine related results obtained by Heyman (2014) and Hu (2014).