Singh expects precipitation between June and September to be between 90 and 96 percent of the long-term average.
India's farm sector accounts for 14 percent of India's nearly $2 trillion economy, with two-thirds of its 1.2 billion population living in rural areas.
Rains are vital to rejuvenate an economy battling its longest economic slowdown since the 1980s and to cool inflation running close to double digits. The new government on Monday made fighting food inflation its top priority.
"The southwest monsoon this year may be less than normal," said Singh, adding that rainfall in July was seen at 93 percent of the long-term average, rising to 96 percent August.
The four-month-long monsoon season started on a weak note as the annual rains arrived over India's southern coast about five days behind the average date of June 1.
India's rice and cotton growing areas are expected to receive deficient monsoon rains, while its soybean growing central region should receive below-average rainfall.
The India Meteorology Department (IMD) defines average, or normal, rainfall as between 96 percent and 104 percent of a 50-year average of 89 cm for the entire season.
"Rainfalls in the first month of the season are expected to be deficient as a result of the weak start of the season," IMD chief L.S. Rathore said, adding that monsoon rains had been 44 percent below average so far.
Half of India's farmland still lacks access to irrigation making many farmers particularly dependent on monsoon rains. The country plans to expand irrigation coverage by at least a tenth by 2017 to cut its dependence on the seasonal rains.
In April, before the start of the monsoon season, the IMD had forecast below-average rainfall in 2014 due to an emerging El Nino in which warm water rises to the surface of the Pacific Ocean.
However, the impact of the El Nino weather phenomenon - which can cause drought in South Asia - is likely to be weak in India, a weather office official said.
"No impact of El Nino is right now seen on the Indian monsoon as it is still in a neutral condition," said D.S. Pai, lead forecaster at the IMD.
India will consider providing subsidised diesel, cheaper loans and extra seeds to farmers if rains are poor this year, Farm Minister Radha Mohan Singh said.
FOOD PRICE INFLATION
The weather office expects the monsoon to cover the whole country by mid-July despite its weak start.
"We have to still wait to see how the monsoon shapes up over the main crop areas as a weak start doesn't necessarily mean a poor monsoon," said Sudhir Panwar, president of Kishan Jagriti Manch, a farmers' lobby group.
Poor rains this year could hit summer crops such as rice, soybean, corn and cotton, raising food prices and pressuring economic growth that has nearly halved to below 5 percent in the past two years, traders said.
"Cotton, soybean and pulses could be hit due to lower rainfall in central and western India," said Harish Galipelli, vice-president research at Inditrade Derivatives and Commodities.
"These crops are mainly grown on rainfed areas. So fluctuation in rainfall can lower their yields. Edible oil imports could rise if soybean production goes down in Madhya Pradesh and Maharashtra."
Galipelli added that rice was not a concern because it is grown across the country, and a drop in production in one state could be compensated elsewhere. There were also enough stocks left over from last year's harvest.