"We will observe them and we will judge," he said.
"If you change your attitude, we will change our attitude."
In the speech, which was carried live by Iranian television, he said Iran was yet to see such a change.
"What is the change in your policy?" he asked.
"Did you remove the sanctions? Did you stop supporting the Zionist regime? Tell us what you have changed. Change only in words is not enough."
Sadeq Saba analyses Iranian response'
The BBC's Sadeq Saba says the Iranian supreme leader may be acting to prevent any internal division between moderates and hardliners over how to react to President Obama's offer.
The demand for the US to withdraw support for Israel is clearly unrealistic, he says.
But any easing of bilateral sanctions or a freeze on Iranian assets could signal to Iran that the US is serious, he adds.
Mr Obama's offer came in a direct video address to mark the Iranian New Year.
In the message, Mr Obama said he was seeking engagement with Iran that was "honest and based on mutual respect".
Obama reaches out to Iran
"My administration is now committed to diplomacy that addresses the full range of issues before us," he said.
The address was distributed to news outlets in Iran with subtitles in Farsi, and posted on the White House website.
Mr Obama's approach - prefigured in his inauguration speech - was seen as a clear departure from the approach of the Bush administration, which described Iran as part of the "axis of evil".
In another possible move towards engagement, the state department is said to be considering an overture in the form of a letter to Iran's Supreme Leader Ayatollah Khamenei before the Iranian elections this summer.
Even so, a week ago Mr Obama extended sanctions against Iran for one year, saying it continued to pose a threat to US national security.
The US fears Iran's uranium enrichment programme is a cover to build atomic weapons, a charge Iranian officials deny.
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