More than 5,000 properties have been affected after an underground cable was damaged, according to Northern Powergrid.
However, City of York Council said that 3,500 properties in the city centre were affected.
After earlier saying that power could be restored by 4.45pm this afternoon, it is now thought that this will not happen until 4am tomorrow.
Numerous traffic light signals were not working, including those at the busy Micklegate Bar, it was believed.
The council is encouraging people to avoid all non-essential travel.
York Station has now re-opened after it was closed to stopping trains this afternoon - but Northern Rail warned that it would be very busy.
North Yorkshire Fire and Rescue Service said that crews from York and Huntington attended a fire involving electrical cables in Toft Green shortly after 3pm.
However the force did not say whether the fire, which was out when crews arrived, was linked to the power cut.
Reports began to surface at around 3.45pm.
Northern Powergrid said that the "unexpected power cut" has affected 5,058 properties in the area.
It later added: "Once our engineers started to fix the power cut we found that an underground cable needed to be repaired following 3rd party damage."
Neil Ferris, City of York Council’s Corporate Director of Economy and Place, said: “An electrical power cut is currently affecting 3,500 properties in central York, as well as West Offices.
“The area affected is from Tadcaster Road/The Mount in the south, down towards the river in the north, across to the railway station and Leeman Road in the west and towards Nunnery Lane in the east.
“West Offices has been closed and all non-essential services and meetings are cancelled. We closed the contact centre at 4.30pm due to low emergency power levels and the safety of those in the building.
“Transport is affected both in terms of street lights, some traffic lights being out and the station is closed. The council’s operation services and depot are not affected.
“Residents and business visitors are asked to check www.york.gov.uk before setting off to West Offices tomorrow. All existing emergency telephone lines remain open.
“We will issue updates on the situation at www.york.gov.uk/poweroutage.”
Bus company First York earlier said that the whole city centre will be experiencing traffic congestion.
An event due to be held at the offices by One Adoption, about adoption services in the region, has been cancelled.