The talks are expected to be dominated by issues of security and trade.
They come days after the US exempted India from economic sanctions in return for cutting imports of Iranian oil.
Ms Clinton said ahead of the talks that ties with India "have never been stronger".
Mr Krishna is being accompanied by a large team, including senior cabinet ministers, government officials and the chief of India's intelligence bureau.
The two sides will discuss a range of issues from Afghanistan and Pakistan to counter-terrorism, homeland security, health and climate change.
BBC India correspondent Sanjoy Majumder says that defence is now a key area of cooperation, with India spending $8bn on US-made military hardware.
Trade has grown from $9bn in 1995 to $100bn this year.
Over the past few months, a number of senior members of the US administration have visited Delhi, including Mrs Clinton and Defence Secretary Leon Panetta.
Treasury Secretary Tim Geithner is expected in the Indian capital at the end of June.
Many in Delhi see this as an indication of the growing significance of India in the US's regional strategy, our correspondent says.
There are areas of concern - the US wants India to allow greater access to American firms, and a landmark nuclear deal has not delivered the lucrative contracts it promised to.
But ahead of the talks, the US announced that it was exempting India from financial sanctions because it had reduced oil imports from Iran, another sign of the growing proximity of the two countries.
Indian Foreign Minister SM Krishna and US Secretary of State Hillary Clinton are due to begin talks in Washington as part of the US-India strategic dialogue.