The interfacial properties near the roughening transition temperature of the modified discrete Gaussian model in two dimensions are discussed. The interfacial profile is calculated by extending the renormalization group method of Kosterlitz and José et al. The square of the interface width and the inverse of the gradient of the profile at the center are found to be given by ln ξ and (ln ξ)1/2, respectively, where ξ is the correlation length of fluctuations. These results are different from those of the mean field theory, the low temperature series expansion and those of Müller-Krumbhaar. The cross-over to the mean field results is also discussed. The interfacial profile is obtained and is expressed by the error function. The present results are applicable to the neutral Coulomb gas with a logarithmic interaction and to the classical X-Y model in two dimensions. Therefore the dielectric function near the metal-insulator transition and the spin correlation function above the transition point are also calculated.