The Paks nuclear plant, about 90km south of the capital Budapest, is operating at less than half its normal capacity, producing 965 megawatts of power rather than the usual 2000.
The plant uses water from the Danube to cool its reactors, but the exceptionally low water level has tested the ability of pumps to bring sufficient quantities into the plant.
Speaking at a news conference at the plant on Friday, Magyar said further reductions in output were imminent later that day, and the entire plant would likely be powered down by Tuesday or Wednesday.
He stressed the shutdown could be performed safely and posed no risk to the public or environment.
The Soviet-built, four-reactor plant at Paks accounts for nearly half of Hungary's electricity production.
If it is powered down, it will be the first time in its 44 years of service.
Hungary and much of Central Europe has been in a prolonged drought in recent months, disrupting tourism, agriculture and industry and confronting residents with the realities of climate change.
Water levels are not expected to rise on the Danube in the coming days or even weeks as no significant rainfall is forecast.
Magyar said the Danube's water level had risen on only three occasions in the month of August in the past 20 years, and added the level at Paks could recede to as low as negative 144cm - far below the previous record of negative 98cm in 2018.
A level of negative 134cm required the full powering down of the plant, he said.
A negative reading means the river has fallen below the gauge's zero reference level, not that the river is empty.
Hungary's government has requested that large companies that consume large quantities of energy voluntarily reduce their consumption to help stabilise the energy supply, particularly in peak demand hours at night.
