Presidential spokesperson Mac Maharaj, who extended the invitation to Shapiro, had said a debate would help "create an environment of dialogue rather than confrontation", the Star reported.
However, Maharaj apparently denied that the presidency extended an invitation to Zapiro for a debate. This was according to a tweet from Beeld reporter, Pauli van Wyk.
On Monday, the suit against Shapiro and the Sunday Times over the "Lady Justice" cartoon was formally withdrawn.
Maharaj said: "Let's look at [the cartoon] through the multicultural lens of South Africa... in an atmosphere where we can listen to each other and stop pretending that any one individual has the infallible answers to all our problems."
He had labelled Shapiro's cartoon racist, and had said it was informed by "ingrained prejudices" about African male sexuality.
Shapiro told the newspaper he considered this a deliberate "red herring", but found the idea of a public debate intriguing.
"I'm never one to shy away from debate, but in this context we have won a total victory (ie the withdrawal of charges) and they're trying to score points now."
Charges were laid against Shapiro and the Sunday Times in 2008 after it published the cartoon depicting Zuma loosening his trousers and apparently about to rape "Lady Justice".
Zuma initially wanted R5m, but the amount was later reduced to R4m. He subsequently dropped the charge of impairment of dignity and further reduced the amount to R100 000 for defamation and an unconditional apology, before dropping the case entirely.