All trains due into the station have been stopped, with five trains stranded and unable to reach platforms.
Network Rail says rapid response teams are working to help passengers get off the trains.
New Street Station has said no one will be able to enter the station until further notice.
The fault causing the problems at the Midlands' busiest station is with the 25,000 volt overhead lines which power the trains.
The rail company said all power has been switched off to the overhead cables.
Network Rail’s Central route director, Denise Wetton, said “as predicted the extreme temperatures have impacted our overhead cables which power trains and we’ve had to stop all trains at Birmingham New Street station.”
“Please keep following our ‘do not travel’ message today as journeys are being severely impacted. We’re sorry to people impacted and we’re working as fast as we can to get things back up and running.”
Rail tickets are currently being accepted on local buses across the West Midlands.
Trainline Avanti West Coast says passengers can use tickets for today (July 19) tomorrow and Thursday.
The operator also announced its services will no longer run for the rest of today and is advising passengers not to travel or come to the station.
Other rail operators such as East Midlands Railway and Cross Country are urging people not to travel to avoid delays and cancellations due to the extreme heat.
Passengers at New Street Station have had trains cancelled and journeys delayed due to the overhead wires.
Meanwhile, West Midlands Ambulance Service attended calls to New Street Train Station following reports of passengers stranded on three trains.
Paramedic and HART officers helped evacuate passengers from the trains onto platform level.