The Congress' decision to later suspend Mr Aiyar and seek his explanation is seen as an attempt to insulate the party from the controversy that erupted just two days before the first phase of Assembly elections in Gujarat.
The BJP and PM Modi, who was campaigning in Gujarat, had invoked Mr Aiyar's remarks to target the Congress through the day.
"Shriman Mani Shankar Aiyar today said that Modi is of 'neech jaati' (low caste) and is 'neech' (vile). Is this not an insult to Gujarat? This is an insult to the great legacy of India. This is born of Mughal mentality," he told an election rally in the state's diamond city, Surat.
PM Modi asked people not to respond to the abusive language on social media or elsewhere. "But if you have any anger over the use of such words against me, press the button alongside the lotus symbol on 9th and 14th (Gujarat polling dates)," PM Modi said.
The BJP has been in power in Gujarat for an uninterrupted 22 years and according to opinion polls, is set to return to power. But most surveys released this week suggested that the Congress, which won 60 of the 182-seat assembly in 2012 elections, could improve its tally.
Rahul Gandhi, who is set to take over as president from his mother Sonia Gandhi later this month, has led the party's campaign in Gujarat and is seen to be anxious not to let the row over Mr Aiyar's remarks mar the party's chances.
"This is the Congress leadership's Gandhian philosophy and a sense of respect towards our rivals," Congress communications incharge Rajdeep Surjewala tweeted, questioning if PM Modi would have "courage to do something similar".
Rahul Gandhi, who had made his disapproval of Mr Aiyar's remarks public, had underlined that the Congress leader was not the only one to have crossed the line. The BJP and PM Modi "routinely use filthy language to attack the Congress party", he tweeted.
Mr Aiyar did apologise, as suggested by Rahul Gandhi, for what the Gandhi family loyalist called his poor command of Hindi. "I meant low level when I said 'neech', I don't think he is low born...if it has some other meaning then I apologise," he said in his conditional apology.
Ahead of the 2014 national election too, Mani Shankar Aiyar had derided Narendra Modi as a tea seller who could never be Prime Minister. He had then refused to apologise.
That comment was seen to have contributed significantly to Mr Modi's sweep of the general election. In his campaign, he had offered his credentials as a self-made leader, stressing the fact that he sold tea on trains as a young boy to help support his family and contrasting his modest origins with those of the Congress' Gandhi family, which controls all power in the party.
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