The tampering has been allegedly done by the Returning Officer.
The court has asked the CBI to complete the probe by July 5 and file a report on July 7, a day before the polls are scheduled to be held.
Justice Amrita Sinha passed the order rejecting the State Election Commision (SEC) and the state government's appeal to conduct the inquiry by the state administration.
"For the purpose of maintaining independence and transparency in the process of investigation, the court thinks it fit to direct the CBI to conduct the investigation and file a report before this court," the court said as quoted by Bar and Bench.
West Bengal Chief Minister Mamata Banerjee-led Trinamool Congress (TMC), which is facing massive corruption allegations on various fronts, is accused by the opposition parties of indulging in violence ahead of the panchayat polls.
A number of political workers have lost their lives in pre-poll clashes in various districts of the state, which is heavily dominated by TMC in terms of political landscape since 2009.
Mamata argues her party is not involved in the clashes and the pre-poll scenario is far better than the ones prevailed during the Left Front regime, particularly in 2003.