We compare the frequencies of volcanic eruptions in the past 270 years with the variations of solar activity and summary curve of principal components of the solar background magnetic field (SBMF) derived from the WSO synoptic magnetic maps. Quartile distributions of volcanic eruption frequencies over the four phases of a 11 year cycle (growth, maximum, descent and minimum) reveal higher numbers of eruptions occurring at the maxima of SBMF of southern polarity with some increases at the minima of sunspot numbers. The frequency analysis of volcanic eruptions with Morlet wavelet reveals that the period of 22 years is more pronounced than 11 years. Comparison of the volcanic frequencies with the summary curve of SBMF for 11 cycles after 1868 (excluding 1950-1980 affected by the open nuclear bomb testing) reveals a strong correlation with the coefficient of -0.84 (within a confidence interval of 95%), while for 8 cycles in the early period of 1750-1868 the correlation becomes nearly twice lower. This lower correlation was likely to be caused by the geomagnetic jerk and migration of the Earth's magnetic pole to lower latitudes. The maxima of volcanic eruptions are shown to occur during solar activity cycles with the southern magnetic polarity that, in turn, can be associated with stronger disturbances of the geomagnetic field leading to the eruption number increase. The next anticipated maximum of volcanic eruptions is expected to occur during cycle 26 (2031-2042), when the solar background magnetic field will have a southern magnetic polarity. These volcanic eruptions can contribute to terrestrial cooling during the modern grand solar minimum (GSM) (2020-2053).