Reddy did not name the groups or provide proof, saying "it is not possible to divulge all this information".
But Indian media reports, quoting unnamed security officials, said the Bangladesh-based Harkatul Jihad al-Islami group known as Huji was behind the attack.
Arati Jerath, the political editor of Indian newspaper Daily News and Analysis, told Al Jazeera she thought the police suspected a "Bangladesh militant group which we call Huji".
"The police here suspect that it [Huji] is based in Bangladesh but they have local modules - sleeper modules - all over, particularly in South India and the state of Maharashtra. These are disaffected youth or fanatical groups."
|People gathered at hospitals for news or to |
identify the bodies of relatives [AFP]
Asked if these groups had been behind earlier attacks, including the bombing of a mosque in Hyderabad in May, she said: "There is a suspicion that it is the same network that was involved.
"I think police have found some connections between these because they've traced some of the chemicals used in the bombs to a lab in Nagpur in the state of Maharashtra. The same lab had supplied chemicals used in some of the other attacks."
No one has claimed responsibility for the blasts but Sriprakash Jaiswal, India's interior minister, said they were part of an effort to undermine the city's mixed Hindu-Muslim community.
Bangladesh condemned Saturday's bombings and rejected the accusations that Bangladeshi groups were involved and Pakistan reacted angrily to Reddy's comments, characterising them as an "irresponsible" knee-jerk Indian reaction.
Tasnim Aslam, a Pakistan foreign ministry spokeswoman, said: "They have been making these allegations and nothing ever came out of those allegations and yet they continue maligning Pakistan."
"Only the Indians have this kind of some supernatural powers that as soon as some terrorist act takes place they know how it happened and who is responsible," she added.
Saturday's blasts were the latest deadly bombings in India that have killed hundreds across the country, including in the capital, financial centre and Hinduism's holiest city, in the last two years.
Indian officials say extremists want to foment tension between India's Hindu majority and its Muslim minority.
The argument has been used even when the victims are Muslim, as was the case in the May bombings of a historic mosque in Hyderabad that 11 killed people.
More than 80 per cent of India's 1.1 billion people are Hindu and 13 per cent Muslim. But in Hyderabad, Muslims make up 40 per cent of the 7 million population.