Recent experiments indicate that the number of hydrogen bonds in solutions increases in the critical region of the solvent. This effect can be explained by the density fluctuations near the critical point. A theory of this unusual effect is developed. The theoretical predictions are compared with the experimental data on hydrogen bonding between perfluoro-tert-butanol and dimethyl ether in supercritical sulfur hexafluoride. The theory requires no adjustable parameters and is in excellent agreement with the experimental data. Possible applications of this effect for hydrogen-bonding solutions and polymer mixtures are discussed.