The Brahmaputra river, which flows from the Himalayas into India and then through Bangladesh, has burst its banks, swamping more than 1,800 villages in the state.
"10 people have died in separate incidents of drowning in the past three days and more than one million people (have been) affected, with the flood situation turning grave," a state government flood bulletin said.
Torrential rains have affected at least 25 of Assam's 32 districts and the federal water resources body said water levels in the Brahmaputra were expected to rise, with more rains forecast over the next three days.
"The flood situation remains extremely critical," Assam's Water Resources Minister Keshab Mahanta told Reuters.
Assam, known for its fine quality teas, has for several years battled floods and heavy rains that have claimed dozens of lives and displaced hundreds of thousands of people.
Assam was on a maximum alert due to heavy rains forecast over the next few days, government officials said.
People in the state were also at threat from Japanese encephalitis, a mosquito-borne viral infection, government officials said. More than 160 people have been killed since January by the infection.
The federal government said it had met on Saturday to review flooding across different parts of the country, and emergency teams had rescued 750 people in Assam and nearby Bihar state, where the Kosi River had risen above the danger mark.Flash floods and landslides in Nepal also triggered by monsoon rains killed 15 people and injured 12 overnight.