A theory of spin fluctuations of excitons in quantum wells in the presence of nonresonant excitation has been developed. Both bright and dark excitonic states have been taken into account. The effect of a magnetic field applied in a quantum-well plane has been analyzed in detail. We demonstrate that in relatively small fields, the spin-noise spectrum consists of a single peak centered at a zero frequency, while an increase of magnetic field results in the formation of the second peak in the spectrum, owing to an interplay of the Larmor effect of the magnetic field and the exchange interaction between electrons and holes forming excitons. Experimental possibilities to observe the exciton spin noise are discussed, particularly by means of ultrafast spin-noise spectroscopy. We show that the fluctuation spectra contain, in addition to individual contributions of electrons and holes, information about the correlation of their spins.