Speaking exclusively to Dutch newspaper de Volkskrant, Blatter claimed the warning, in which Platini allegedly said Blatter would go to jail if he did not withdraw from May's Fifa presidential race, was made to the Swiss official's brother.
But a source close to Platini, contacted by the AFP news agency, described the claim as "ridiculous".
Blatter said he found his 80-year-old brother Peter crying at the Fifa congress in Zurich in May, where the 79-year-old Blatter was re-elected for a fifth Fifa term, before later vowing to step down and hold new elections.
"During lunch, Platini had sat down at my brother's table and said: ‘Tell Sepp to withdraw from the election or he will go to prison'," said Blatter.
Blatter claimed the exchange happened before the elections but his brother only told him about it afterwards.
"This complete fabrication is the latest in a series of attempts from Zurich to distract everyone from the problems Fifa is confronted with," a source close to Platini told AFP.
"The UEFA president will not dignify these ridiculous allegations with a response."
Blatter continued his claims by saying he felt betrayed by the Frenchman, 60, who is the favourite to succeed the Swiss in February's new election.
"You know, there was a time when our relationship was like that of a father and his son," said Blatter.
"He worked for me for four years at Fifa, after the 1998 World Cup in France. Together we prepared his board memberships of both Uefa and Fifa. In 2007, he also became president of Uefa, with my direct support."
Blatter said Platini has "changed" but that he does not know why the Uefa chief, a former France international player and coach, should have turned against him.
"You will have to ask him about his character. I don't know what goes on in his head," said Blatter.
Although Blatter was re-elected, despite a corruption scandal engulfing the whole process after seven Fifa officials were arrested over accusations of bribery, the Swiss announced four days later that he would resign.
"The mounting pressure on me and Fifa left me no other choice. I was faced with a tsunami," said Blatter.
"It was a shock. It was not an easy decision to take, but it gave me time to think about what had gone wrong.
The source close to Platini said the Frenchman is concentrating on rehabilitating Fifa's reputation following the corruption scandal rather than responding to allegations from Blatter.
"Michel Platini is currently more concerned with the preparation... of a programme to restore the image and reputation of Fifa," said the source.