A Maharashtra ATS official said the initial assessment suggests that the account does not appear genuine but the investigators are not taking the matter lightly.
"All the officers of the ATS have been instructed to be alert and city police has been asked to ensure tight security," the official said.
Hours after Sunday's Bodh Gaya serial blasts, a tweet by @IndianMujahidin - which claims to be the IM's official Twitter account - said: "9 Dhamake Humne Karay (We carried out the nine blasts)".
The tweet about Mumbai -dated July 5, two days before the Bihar blasts - reads, "Hamara Agla Target Mumbai Hai. Rock Sago Toh Rock Lo, 7 days Left (Our next target is Mumbai. Stop us if you can. Seven days left)."
The National Investigation Agency (NIA) is also probing the authenticity of the account.
Mumbai police has sensitized its personnel and tightened the security across the metropolis.
Joint Police Commissioner (Law and Order) Sadanand Date refused to respond to the twitter post but added, "Going by the current situation, we have sensitised our policemen and appropriate steps are being taken to ensure security at vulnerable places. We have also alerted the security men at vital locations."
"We appeal to the citizens to be alert and inform the policemen in case they find any suspicious object," Date told PTI.
Nakabandis were being conducted and patrolling had been intensified at strategic locations, police said, adding that frisking at the suburban railway stations was being done and dog squads were put on service.
"Vehicles are being stopped randomly for checking," said a police official.
Railway compartments were being checked and announcements made to report to police in case passengers found any suspicious object, a railway policeman said.