The amber, which was found in Bexhill, was formed about 140 million years ago in the Cretaceous period.
Amateur fossil hunter Jamie Hiscocks and his brother Jonathan also found the fossilised remains of an Iguanodon jaw bone on the coastline.
The spider web thread is now being studied at Oxford University.
Trapped in resin
Professor Martin Brasier, a palaeobiologist at the university, said: "You can see where the web is attached to the surface.
"If it is confirmed - and we think we have got good evidence for it - then it would be the oldest preserved spider's web and the oldest fossil silk, I think, in the fossil record."
Fossilised charcoal was also found in the fossil beds near to the amber at low tide on the coast between Hastings and Cooden.
Scientists think the web became trapped in conifer resin after a forest fire and then became fossilised inside the resulting amber.
Jamie Hiscocks, who has been hunting for fossils for more than 10 years, said: "The strata beds are actually being broken up all the time.
"Most of the pieces are actually just lying there on the surface. You pick it up, just bend down and pick it up."