We present the stellar population and velocity dispersion gradients for a sample of 24 brightest cluster galaxies (BCGs) in the nearby Universe for which we have obtained high quality long-slit spectra at the Gemini telescopes. With the aim of studying the possible connection between the formation of the BCGs and their host clusters, we explore the relations between the stellar population gradients and properties of the host clusters as well as the possible connections between the stellar population gradients and other properties of the galaxies. We find mean stellar population gradients (negative {Δ}[Z/H]/log r gradient of -0.285{±}0.064; small positive {Δ}log (age)/log r gradient of 0.069{±}0.049; and null {Δ}[E/Fe]/log r gradient of -0.008{±}0.032) that are consistent with those of normal massive elliptical galaxies. However, we find a trend between metallicity gradients and velocity dispersion (with a negative slope of -1.616{±}0.539) that is not found for the most massive ellipticals. Furthermore, we find trends between the metallicity gradients and K-band luminosities (with a slope of 0.173{±}0.081) as well as the distance from the BCG to the X-ray peak of the host cluster (with a slope of -7.546{±}2.752). The latter indicates a possible relation between the formation of the cluster and that of the central galaxy. ; Comment: 23 pages, 18 figures, accepted for publication in MNRAS. arXiv admin note: text overlap with arXiv:1104.2376v1