Contractor Alun Griffiths was edging the 4,260t replacement bridge into place when the two Osprey self-propelled transporters sank into the mud 30m short of their final position.
The contractor is now racing against time to solve the problem so it can reinstate rail journeys from Bristol to Wales which were disrupted when the original Gipsy Patch Lane rail bridge was demolished to make way for the weekend bridge replacement.
Two dozers were drafted in to try to shunt the transporters forward but these are understood to have failed to budge it.
Network Rail had said prior to the weekend it was due to reopen next Saturday, but the disruption is now expected to continue for longer than originally anticipated.
A South Gloucestershire Council spokesman said: “Despite detailed planning and preparation, our contractor, Network Rail, has encountered a significant issue moving the new 4,260-tonne Gipsy Patch Lane bridge into place.
“The bridge, which is already built, is being moved into position on a number of independent hydraulic transporters.
“Having successfully moved two-thirds of the way into place the bridge transporters are now stuck.
“Our focus, alongside Network Rail, is on resolving the issue so that the bridge can be moved into position. This is likely to take a number of days and will delay the reopening of the mainline.”