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One eyewitness told the BBC he had seen a gunman opening fire in the Taj Mahal's lobby.
He said he had seen people fall before he fled the lobby.
"All I saw was one man on foot carrying a machine gun type of weapon - which I then saw him firing from and I saw people hitting the floor, people right next to me," he said.
BOMB ATTACKS IN INDIA IN 2008
30 October: Explosions kill at least 64 in north-eastern Assam
30 September: Blasts in western India kill at least seven
27 September: Bomb blasts kills one in Delhi
13 September: Five bomb blasts kill 18 in Delhi
26 July: At least 22 small bombs kill 49 in Ahmedabad
25 July: Seven bombs go off in Bangalore killing two people
13 May: Seven bomb hit markets and crowded streets in Jaipur killing 63
International reaction
There has been a wave of bombings in Indian cities in recent months which has left scores of people dead. Most of the attacks have been blamed on Muslim militants, although police have also arrested suspected Hindu extremists.
Mumbai itself has also been attacked in the past: in July 2006 a series of bomb attacks on busy commuter trains killed almost 190 people and injured more than 700.
Police accused Pakistan's intelligence agency of planning those attacks, which they said were carried out by an Islamist militant group, Lashkar-e-Toiba.
Pakistan rejected the allegation, saying there was no evidence that its intelligence staff were involved.
But the latest shootings come at a time when ties between India and Pakistan have improved.
Just days ago Pakistani President Asif Ali Zardari told a summit in Delhi that Pakistan would not be first to carry out a missile strike on India.
The two countries have a joint anti-terror mechanism whereby they are supposed to share information on terrorist attacks.
