Observations show that twisted magnetic flux tubes are present throughout the Sun’s atmosphere. The main aim of this work is to obtain the damping rate of sausage modes in the presence of weak magnetic twist. Using the connection formulae obtained by Sakurai et al., we investigate resonant absorption of the sausage modes in the slow continuum under photospheric conditions. We derive the dispersion relation and solve it numerically, and consequently obtain the frequencies and damping rates of the slow surface sausage modes. We conclude that the magnetic twist can result in strong damping in comparison with the untwisted case.