Mahmoud Ahmadinejad made the offer to George Bush on Tuesday. Thursday is the deadline set by the UN Security Council for Iran to suspend all uranium enrichment and reprocessing activities, and Iran faces possible sanctions if it fails to comply.
Ahmadinejad said: "I suggest we talk with Mr Bush, the president of the United States, in a live television debate about world issues and ways out of these standoffs.
"We would voice our opinions and they would too. The debate should be uncensored, above all for the American public."
The White House called the suggestion a "diversion" from the Thursday deadline and refused the invitation.
Dana Perino, the White House spokeswoman, said: "Talk of a debate is just a diversion from the legitimate concerns that the international community, not just the US, has about Iran's behaviour, from support for terrorism to pursuit of a nuclear weapons capability."
Earlier this year Ahmadinejad sent Bush a letter, the first contact in decades between leaders of the two countries.
But the Iranian president said that such a debate would not necessarily mean reopening dialogue with the United States, which froze diplomatic relations with Iran after the seizure of its embassy in Tehran in 1979.
Ahmadinejad said: "Debate is different from dialogue, dialogue has other conditions, we have said our position on that before."
But he said that dialogue was also possible with "the ones who show a frown to our nations if the conditions are fulfilled".
The Vienna-based International Atomic Energy Agency is to report to the Security Council, also on Thursday, on Iran's compliance with its demands.
Iran has said repeatedly that it has no intention of abandoning its nuclear work, which it says is for civilian energy purposes only.
Ahmadinejad said he believed that it was "unlikely" the Security Council would act against Iran over its nuclear programme, which the United States sees as a cover for weapons development.
John Bolton, the American ambassador to the UN, has said the United States plans to put forward a draft resolution imposing penalties such as a travel ban and asset freeze for key Iranian leaders soon after the deadline.
Ahmadinejad said: "Sanctions are not an issue ... We will not be happy if they use anything but logic but we are not worried. After all, we are capable of defending our rights."
The Iranian president caused controversy last year when he described Israel as a tumour that should be "wiped off the map" and said he wanted the root of tensions in the Middle East to be "removed".