Workers at the yard suspended strike action to talk with dockyard owners Babcock Marine.
Heathcliffe Pettifer from Unite the Union said a joint statement had been released by the company, GMB and his union.
He said it stated: "The company and the trade unions have had positive discussions in order to reach a settlement to the pay agreement and industrial dispute.
"Significant progress has been made towards the company tabling a revised offer and discussions are to continue on Thursday.
"Industrial action remains suspended to enable a conclusion of these discussions in a situation of normal work."
Some 180 workers took part in strikes over a two-year pay deal.
The action was suspended last week after shipyard owners Babcock Marine said work earmarked for Appledore could go elsewhere if the dispute was not resolved by the end of the year.
Mr Pettifer added: "I believe we are close to it being resolved. The meeting on Thursday will be between the local stewards and the local management because ultimately it is up to them to find a resolution."
The workers are campaigning for equal pay rates to those of other yards.
The deal concerns a two year contract for an aircraft carrier, which will secure at least 200 jobs in Appledore.