We consider the following extension of the concept of adjacent strong edge colourings of graphs without isolated edges. Two distinct vertices which are at distance at most r in a graph are called r-adjacent. The least number of colours in a proper edge colouring of a graph G such that the sets of colours met by any r-adjacent vertices in G are distinct is called the r-adjacent strong chromatic index of G and denoted by χa,r′(G). It has been conjectured that χa,1′(G)≤Δ+2 if G is connected of maximum degree Δ≥2 and non-isomorphic to C5, while Hatami proved that there is a constant C, C≤300, such that χa,1′(G)≤Δ+C if Δ>1020 [J. Combin. Theory Ser. B 95 (2005) 246–256]. We conjecture that a similar statement should hold for any r, i.e., that for each positive integer r there exist constants δ0 and C such that χa,r′(G)≤Δ+C for every graph without an isolated edge and with minimum degree δ≥δ0, and argue that a lower bound on δ is unavoidable in such a case (for r>2). Using the probabilistic method we prove such an upper bound to hold for graphs with δ≥ϵΔ, for every r and any fixed ε∈(0,1], i.e., in particular for regular graphs. We also support the conjecture by proving an upper bound χa,r′(G)≤(1+o(1))Δ for graphs with δ≥r+2.