Water flow at the dam has surged over the warning levels of 45,000 cubic meters per second and is expected to reach 60,000 cubic meters per second by Wednesday, the Xinhua news agency said.
The massive dam is in Hubei province, to the east of Sichuan province and Chongqing municipality where torrential rains have left at least 105 people dead and 83 missing in landslides and floods.
The governments of Chongqing and Hubei, which are situated along the upper and middle reaches of the Yangtze, have put all departments on high alert, Xinhua said.
"The flood will challenge embankments in the middle and lower reaches of the Yangtze River and may cause mountain torrents, landslide and mud-rock flows ...," said Chen Qijing, an official with the Hubei provincial flood control headquarters.
"We've sent people to keep a 24-hour watch at local embankments and have stepped up publicity work to raise the local people's awareness of the flood."
The water level at Yichang hydrological station in Hubei is expected to reach 53.3 meters (175 feet), 0.3 meters above the warning level, by Wednesday evening.
The controversial and expensive Three Gorges Dam, meant to tame the flood-prone Yangtze and increase the country's power supply, is the largest hydroelectric project in the world.