According to [\textit{A. George} and \textit{P. Veeramani}, Fuzzy Sets Syst. 64, 395--399 (1994; Zbl 0843.54014)], a fuzzy metric space is a triple \((X,M,*)\) where \(X\) is a set, \(*\) is a continuous \(t\)-norm and the fuzzy metric \(M\) is a mapping \(M: X\times X\times (0,\infty) \to [0,1]\) satisfying certain conditions. In the same paper the authors showed how this metric induces a topology \(\tau_M\) on the set \(X\). The authors of this paper define a fuzzy metric \(H_M\) on the set \(K(X)\) of all nonempty compact subsets of \((X,\tau_M)\), that is actually a fuzzy set \(H_M: K(X)\times K(X)\times (0,\infty) \to [0,1]\) which is a fuzzy analogue of the Hausdorff-Bourbaki metric, and study some properties of the corresponding space: such as completeness, completion and precompactness. A typical result states that the space is \((X,M,*)\) is precompact iff the space \((K(X),H_M,*)\) is precompact. Some examples are given. Possible applications of the obtained results are discussed.