Delhi has been targeted by militants a number of times in the past
The US has issued a travel alert saying that there are "increased indications" that "terrorists" are planning attacks in the Indian capital, Delhi.
The alert listed some popular shopping areas and advised Americans travelling or residing in India to be vigilant.
Delhi police chief YS Dadwal told Reuters that "threats are there", and the police were "wide awake" to it.
The alert came after bombs exploded at the weekend outside a cricket stadium in the southern city of Bangalore.
The US advisory mentioned Chandni Chowk, Connaught Place, Greater Kailash, Karol Bagh and Mehrauli as places which are "attractive targets for terrorist groups".
Australia has also put out an advisory warning tourists to avoid the same areas.
Most of these places contain busy, crowded markets, which are popular with locals and foreigners alike.
Delhi police chief YS Dadwal said security was the "top priority".
"I cannot pre-empt threats, threats are there, but we are wide awake," Mr Dadwal said.
"We will not allow anything to happen, security is top priority."
There have been concerns about security in the Indian capital ahead of the upcoming Commonwealth Games - the event, involving 54 nations, will be the biggest international sporting event ever hosted by India.
Delhi has been targeted before by militant groups.
In 2008, 20 people were killed and 90 wounded when blasts ripped through busy shopping areas.
And in 2005, more than 60 people were killed and 210 injured in two explosions in the crowded Sarojini Nagar and Paharganj markets.