Passengers on board trains reported being told the bird was blocking part of the route.
Transport for London’s (TfL) website stated there are no services between Heathrow Airport in west London and Abbey Wood in south-east London – via the centre of the capital – due to “an obstruction on the track”.
It added that there are “severe delays” on the rest of the line.
National Rail Enquiries said there are “animals on the railway”.
Passengers reported being held on trains – which do not have toilets – for up to hour due to a swan on the tracks at Paddington.
One Twitter user wrote: “I have been trapped on the Elizabeth line for an hour because of a swan on the line.”
Another posted: “You are an absolute shambles @TfL. I cannot believe you closed the entire Elizabeth line for a swan on the tracks.
“Has nobody been on the shooing course. How hard is it to move a swan?”
The Elizabeth line stretches from Reading in Berkshire and Heathrow Airport to Shenfield in Essex and Abbey Wood.
It opened in May 2022 at an estimated cost of £18.9 billion.