Shinde said he had only stated what has been appearing in the media and his reference was to "saffron terrorism".
Speaking at the All India Congress Committee's Chintan Shivir in Jaipur, Shinde referred to the involvement of right-wing elements in terror incidents and said there was a need to be alert against misinformation.
"After investigations, we have seen that be it the BJP (Bharatiya Janata Party) or the RSS (Rashtriya Swayamsevak Sangh), their training camps are promoting Hindu terrorism. This is a matter of concern. We are keeping a strict eye on it," Shinde said.
"There was nothing new. It has come in the papers so many times. I have spoken of saffron terrorism," he told reporters later.
In his speech, Shinde referred to blasts in the Samjhauta Express, at the Mecca Masjid in Hyderabad and in Malegaon in Maharashtra and said there was misinformation about bombs being planted by members of the minority community.
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BJP leader Sushma Swaraj said: "Saffron is the antonym of terrorism. Saffron is the symbol of our traditions, culture, renunciation and sacrifices. The home minister must apologise to the nation."
BJP vice-president Mukhtar Abbas Naqvi said Shinde's remarks were "objectionable, dangerous and unfortunate" and demanded an apology from Congress president Sonia Gandhi and general secretary Rahul Gandhi, now the party vice president, apart from Shinde.
Naqvi said a nationalist organisation has been linked to terrorism and Shinde's remarks amounted to "disturbing the peaceful atmosphere for political reasons".
"It is very unfortunate. Terrorists are feeling encouraged. They are infiltrating and carrying out their activities. The instrument to fight terrorists has been repealed. But the party is silent on it," Naqvi said.
Referring to the brutal killing of two Indian Army soldiers on the Line of Control (LoC) in Jammu and Kashmir earlier this month, Naqvi said militants could be involved in the incident.
"The prime minister reacted late on the incident. The government gives warnings in compulsion. The Congress has been silent on it," he said.
He termed the Congress a "defaulter party" and said the government it leads is "defamed". He also attacked the party on corruption and inflation.
"With the blackness of coal, a garland of price rise and ankle bells of scams, what kind of a dance is this?" Naqvi said.
Congress leaders spoke in different veins over Shinde's remarks with spokesperson Rashid Alvi stating that the party did not see terrorism linked to any religion.
Alvi said that whatever Shinde said would have been based on documents.
However, he added that terrorism did not have a colour or was linked to a religion.
"The Congress does not see terrorism as linked to any religion. Terrorism is terrorism," he said.
Minister of State for Parlianmentary Affairs Rajeev Shukla said there was "no Hindu or Muslim terrorism."
"What he meant was right wing terrorism," Shukla said.
Congress leader Mani Shankar Aiyar told reporters: "We have known this for long. Shinde had the courage to say it."
Party leader Digvijaya Singh said there was no Hindu terrorism but there was "Sanghi terrorism".
"It (terrorism) is not linked to a religion but is linked to an ideology," Digvijaya Singh said.