"Inside the walled historic city a series of blasts went off starting at around 7:30 at night and lasting about half an hour.
"We are hearing reports from eyewitnesses on the ground of carnage. One bomb is a suspected car bomb.
"There are no claims of responsibility as yet. There's lots of speculation that these could be fighters that have infiltrated from Pakistan," he said.
McClure added that one bomb went off outside a Hindu temple, suggesting that there may be communal origins behind the blasts.
The historic city in Rajasthan state is a popular destination for foreign and Indian tourists.
The walled city is full of tourist sites and bazaars.
Vasundhara Raje, chief minister of the desert state of Rajasthan, said: "We will not tolerate such activities at all."
Manmohan Singh, the Indian prime minister, condemned the blasts and appealed for calm.
Alerts have been given for New Delhi, the capital, and Mumbai, India's main financial centre.