A number of biporous granulated acid faujasite-type zeolite-containing catalysts based on Ukrainian kaolin were in situ synthesized. The samples were characterized using XRF, XRD, FTIR, ammonia TPD, pyridine adsorption with IR control, and low-temperature nitrogen adsorption/desorption. The samples possess non-identical micro- and mesoporosity as well as weak, medium, and strong acid sites of different nature (Brønsted and Lewis sites). The activity of composite nanostructured zeolite materials in glucose transformation into 5-hydroxymethylfurfural was determined. A range of studies at different temperatures and reaction times using factor planning methods were conducted on the best catalyst. A full factorial experiment with two independent variables at two levels (temperature and reaction time) was implemented. According to the results of experimental data processing, the regression equation in the coded values of variables has been obtained. Response surfaces for the yields of 5-hydroxymethylfurfural and light reaction products were built. The influence of peculiarities of porosity and strength of zeolite acid sites on the activity and selectivity of the catalysts in 5-hydroxymethylfurfural synthesis is discussed. The selectivity of zeolite-containing catalysts for 5-hydroxymethylfurfural is suggested to increase with an increasing proportion of acid sites of medium strength. Weak acid centers promote the oligomerization of both the original glucose and transformation products, while strong acid sites convert glucose into lighter than 5-hydroxymethylfurfural products. The acidity of the catalysts was found to have a greater impact on the efficiency of large-porous faujasite catalysts than developed mesoporosity. Improved mesoporosity of catalysts enhances better desorption of reaction products, and as a result, the extent of catalyst deactivation decreases.