Art, antiques and collectables more than 50 years old deemed cultural goods must be accompanied by an export licence from the Arts Council if they are to be exported.
The British Art Market Federation (BAMF) and its association members are looking to prevent the current backlog extending any further.
Anthony Browne, chairman of BAMF, said: “We are working with Arts Council England to create an interim solution that will allow the system to begin running again and we have been trialling it this week.
A spokeswoman at the Arts Council said: “We hope to open the ELU as soon as possible, and are exploring ways to do this under the current circumstances.”
Browne added: “In the medium term it is imperative we get a digital system in place as soon as possible. It will be detrimental to the art market if we still have to rely on this paper system by the end of the year when we leave the European Union’s single market and the customs union.”