Mr Aiyar today called PM Modi a "neech aadmi" (vile man) who does "dirty politics" after the prime minister targeted the Congress saying that parties seeking votes in BR Ambedkar's name tried to erase his contribution to nation building the party.
"The strength of India's democracy will be displayed when a person of humble background politically defeats the dynasty and its representatives," the finance minister said in a tweet.
Mr Jaitley also said that the comments were part of a strategy of the Congress to belittle the Prime Minister every day and then say they will "retract".
"If you are unsure of the language or facts, then you are unfit to be in politics," he said referring to Mr Aiyar's defence that Hindi was not his language.
Congress vice-president Rahul Gandhi has asked Mr Aiyar to apologise to the Prime Minister.
BJP and PM routinely use filthy language to attack the Congress party. The Congress has a different culture and heritage. I do not appreciate the tone and language used by Mr Mani Shankar Aiyer to address the PM. Both the Congress and I expect him to apologise for what he said.- Office of RG (@OfficeOfRG) December 7, 2017
During the 2014 general election, Mr Modi was derided by Mani Shankar Aiyar as a tea seller who could never be a Prime Minister. "Narendra Modi will never become the Prime Minister of the country... But if he wants to distribute tea here, we will find a place for him," Mr Aiyar had infamously said.
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.
(with additional inputs from agencies)