A piece of cable came into contact with the overhead wires at approximately 2:45pm on Tuesday, which caused a power failure and damaged the signalling equipment that controls train movements.
Network Rail engineers have been on site working to repair the problem all night.
However, this is complicated equipment and passengers are being warned that disruption may last all day.
Rob McIntosh, route managing director for Network Rail, said: “I am sorry for the disruption this has caused to passengers. Our engineers have been working through the night and new teams are on site today to fix the damage. At this point we cannot say how long that work will take.
“I understand how frustrating this is for passengers and we will provide more information about when full services will resume as soon as possible.”
A Thameslink spokesperson said: “We’d like to apologise for the continued severe disruption so many of our passengers are experiencing. While Network Rail is working on the problem we urge people to check travel advice at thameslinkrailway.co.uk and use alternative transport wherever possible.
“We are doing all we can to keep people updated through our web, app and Twitter feed and have extra staff out on the ground.”
An East Midlands Trains spokesperson said: “We’re sorry for the disruption our customers are experiencing today. We’re working hard to keep customers on the move whilst Network Rail engineers carry out the necessary repairs works.
“We have detailed information on our website which includes advice on alternative routes that our customers can use, and we also have extra staff out at stations helping customers with their travel today.”
Passengers who are delayed by 30 minutes or more should claim compensation via the operators’ websites.
East Midlands Trains are running a reduced service, operating to and from London.
Customers are strongly advised to check before travel at: http://www.eastmidlandstrains.co.uk/information/media/Disruption-Information/DELAYS-Luton-Bedford/
Thameslink are running a limited service of, at most, three trains per hour in operation from Bedford to London St Pancras and there are no cross-London trains beyond St Pancras.
There is also a limited Thameslink service only from Wimbledon and Sutton into London Blackfriars, while Sevenoaks services run to Blackfriars and Victoria only.
Trains from Brighton area are also disrupted. Passengers are urged to use alternative routes wherever possible, including the Great Northern route from stations such as St Neots, Hitchin and Hatfield into London King’s Cross.
Thameslink car park season tickets are valid at Great Northern stations and other Thameslink stations.
Please check: www.thameslinkrailway.com for more information.