Washington has said it will now push for tougher sanctions against Iran over its continuing nuclear activities.
Mr Burns has backed the idea of "additional sanctions"
The United Nations Security Council gave Tehran 60 days to suspend its uranium enrichment programme.
But the deadline has expired and a UN report is soon expected to confirm that Iran is pressing ahead with developing its own nuclear fuel cycle.
US Under Secretary of State Nicholas Burns accused Iran of "brazenly pursuing" its nuclear ambitions.
Mr Burns said that over the next few weeks there would be efforts within the UN Security Council to establish "additional sanctions" on the Iranian Government.
On Wednesday, President Mahmoud Ahmadinejad said Iran would try to achieve nuclear capability as soon as possible, according to reports by Iran's Isna news agency.
However, for the first time a political party in Iran has also called on Mr Ahmadinejad to accept the UN's demands.
Iran denies Western claims that it is seeking nuclear weapons, saying its programme is for purely peaceful ends.
A UN resolution, adopted on 23 December 2006, imposed sanctions against Iran's nuclear and missile programmes and opened the way for further measures if it failed to halt uranium enrichment within two months.
Following the deadline's expiry on Wednesday, the UN's nuclear watchdog, the International Atomic Energy Agency (IAEA), is expected to report that Iran has defied the 60-day ultimatum.
Iran resumed uranium enrichment last year. The process can produce fuel for power stations or, if greatly enriched, material for a nuclear bomb.
Speaking in the northern town of Siahkal, Mr Ahmadinejad said: "Obtaining this technology is very important for our country's development and honour. It is worth it to stop other activities for 10 years and focus only on the nuclear issue."
But one small radical reformist political party, the Islamic Revolutionary Mujahadin Organisation, has complained that Iran's drive to produce nuclear energy has endangered national security, the national interest and the destiny of the Iranian people.
The BBC's Frances Harrison says this is the first time there has been open criticism of Mr Ahmedinejad's nuclear policy. Allies of the president in parliament were quick to say it came from lackeys of the United States who did not even know the basics of politics, our correspondent adds.