Sumit Khambekar, chief of the Mumbai unit of the Maharashtra Navnirman Sena, and social activist Shahzaad Poonawala filed separate complaints against Varma in police stations in the city.
Khambekar told media persons that Varma's tweets "hurt the sentiments of a large number of Hindus, especially devotees of Lord Ganesh".
Poonawala said Varma's tweets have "angered" not only Maharashtrians but also Muslims of the state.
The 11-day Ganeshotsav is being celebrated across Maharashtra with glittering decorations on roads and bylanes, rows of fairy lights, and patriotic and social themes.
Varma Friday posted certain tweets and jocular remarks on Lord Ganesh, which many people said were objectionable and in poor taste.
Police said they will investigate the case before taking further steps.
Social activists and organisations, including the Vishwa Hindu Parishad, have condemned Varma's comments and demanded an apology from him as well as police action.
