Confirmed cases of the disease are becoming less frequent
Mr King said that enough time had now elapsed between the last confirmed case of the disease, on 30 September, for him to be hopeful - although not confident - that the epidemic was nearing an end.
However, he warned this was not a signal for UK farmers to relax their disinfection and isolation procedures.
And he stressed that Britain would need to be free of the disease for at least three to four months before government officials could be truly confident.
On Thursday, the EU announced it would allow some British pork exports to resume.
Counties which have not had a case of the disease and which do not adjoin high-risk areas will be able to export pork and bacon from 22 October.
Mr King's comments come as a public inquiry in Devon hears how efforts to tackle the disease in the early stages were hampered by government red tape.
The Royal Society for the Protection of Animal's (RSPCA) South West superintendent, John Tressider, is expected to tell the hearing in Exeter that there was an initial reluctance by the Ministry of Agriculture (Maff) to involve the RSPCA.
Problems outlined
Serious welfare problems caused by delays in issuing movement licences, inadequate supervision of slaughter, and delays in issuing guidelines to slaughter teams, were outlined in a report submitted to the inquiry.
Mr Tressider will say on Friday - the fifth day of the inquiry - that the RSPCA in the west country had to deal with "daunting welfare problems" during the height of the crisis.
"We were dealing with livestock suffering in a way that ought to lead to prosecution," he said.
But the liaison between the RSPCA and the Department for Environment, Food and
Rural Affairs (Defra) - Maff's successor - was now working well, he said.
"We must build on this to develop effective contingency plans for the future, with safeguards for animal welfare at the forefront," he added.
Devon was one of the counties worst affected by the disease, with 173 cases confirmed and 390,000 animals slaughtered. Devon County Council launched its investigation into the crisis in August. It received some 400 submissions from across the community.
A report on the hearings will be sent to one of the government's inquiries, the Policy Commission on the Future of Farming, by the end of the month. The hearings, which are being webcast live on the internet, will end on Friday.