Local television said the surgery on the 76-year-old leader could last six hours. Singh may be unable to return to work for at least several weeks.
An official at the prime minister's office said the operation was expected to have started at 8:30am (0300 GMT). Local television channels said the operation had begun.
Foreign Minister Pranab Mukherjee will take over most prime ministerial responsibilities, including defence and security, while Singh was recovering from the surgery, an official at the prime minister's office said.
Mukherjee will also take over the finance minister portfolio, which had been under Singh.
"Pranab Mukherjee is in charge of the government until the prime minister resumes his duty," the official, who asked to remain anonymous, told Reuters.
"This is normal procedure as he is the most senior member of the cabinet, so there has been no official announcement."
That lack of announcement led to speculation that the ruling Congress party did not want to give Mukherjee a public endorsement as acting prime minister just before an election, and overshadow other candidates within Congress battling to be Singh's successor.
Many ceremonial duties, for example, will be transferred to the vice-president, not Mukherjee.
The operation came just as reports surfaced that Rahul Gandhi, heir to one of India's most powerful family dynasties, was emerging as a potential successor.
A general election is due by May this year and Singh has been expected to continue as prime minister if the Congress coalition wins. The main battle is between the Congress-led government and a coalition led by the Hindu-nationalist Bharatiya Janata Party.
The surgery could mean that Singh will not be able to carry out full vote campaigning. But the soft-spoken economist was never central to Congress's campaigning plans, analysts said.
Singh underwent bypass surgery in 1990 in Britain. He also underwent wrist surgery in 2006, a prostate gland surgery and a cataract removal procedure last year, officials said.
One expert said on Friday a new operation presented little risk for the prime minister.
"There is 1 to 2 per cent risk factor associated with a bypass surgery on a patient with a long history of cardiac problems like the prime minister," said Dr TS Kler, head of cardiology at the Escorts Heart Institute & Research Centre Ltd.
In neighbouring Bhutan, hundreds of Buddhist monks have begun special prayers for the Indian prime minister.