The party said any debate on terrorism should be comprehensive covering "violations" in Kashmir Valley, unfurling of IS flags on Indian soil and talks in Ufa with Pakistan and the Prime Minister should reply to it.
Congress' response came after Home Minister Rajnath Singh, while speaking in the Lok Sabha, accused the previous UPA government of weakening the fight against terror by coining the term "Hindu terrorism".
Raising the issues of unfurling of the flags of other countries and IS on Indian soil and the Prime Minister's "silence", party's senior spokesman Anand Sharma said, "We have strong objection to what is happening. We have serious concerns to share."
He also demanded that Prime Minister Narendra Modi alone should reply to any such discussion in Parliament, saying neither his External Affairs Minister or any other Minister accompanies him in his visits abroad.
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"The debate must be comprehensive, inclusive and it has to be replied to by the Prime Minister because his Home Minister and Foreign Affairs Minister and other Ministers do not accompany him, because he is his own person and that is his choice. Only he knows and his NSA. NSA cannot respond to Parliament," Sharma said.
"We are only left with Pakistani version of what was discussed & what was committed. This country needs to be taken in confidence," he said on the July 10 meeting between Pakistan Prime Minister Nawaz Sharif and Modi in Russia.
Lashing out at Singh, he said that the party does not need lessons on the issue of terrorism from anyone.
In the Lok Sabha, Leader of Congress Mallikarjun Kharge accused the government of dividing Parliament with Singh's "political speech" on terror.
"We all welcomed, clapped (Singh's statement)... Because the country is one and there is no compromise on terrorism and unity and integrity of the country," Kharge said, refuting Singh's charge that the Congress was making it appear that Parliament was divided on terrorism.
He then went on to attack Singh, saying "just to divide this House and to blame the Congress, he said what China did... Was it necessary? He did it just to defame the previous governments. This is not a political statement. You are doing politics. We won't tolerate that. It should be expunged.