"The police has seen it. It also has a few seconds of the bicycle [used in the attacks]."
The video, sent by e-mail, was later broadcast on national television channels in India.
Pankaj Singh, Jaipur city police chief, said the mail bore the sender's name as "Indian Mujahidin", a group about which little is known.
"It's a post-dated e-mail and it was sent after the attacks claiming 'we did it' and we are trying to verify whether it is the source or a false claim," he said.
Authorities imposed a curfew on the city in the wake of the attacks. The curfew was re-imposed for a second day on Thursday.
Over 60 people were killed and hundreds of people were injured in the blasts, which targeted busy markets and Hindu temples in Jaipur, one of India's major tourist destinations.
The city was hit by a series of seven bombs on Tuesday night, with an eighth bomb, which failed to explode, defused by police.
Ashok Parnami, the city's mayor, said Jaipur had been placed in "a state of terror" by the attacks.
Dozens of people have been arrested in connection with the attack and on Wednesday evening, police released a sketch of a man in his mid-20s who they suspected of being involved in the bombings.
Pakistan-based groups fighting against Indian rule in Kashmir are usually blamed for such attacks which have plagued major Indian cities for years.