Hartlepool Council had earlier refused to give planning permission to extend Ables facilities to recycle the US ships. It citied environmental and planning concerns as the reason and Able had been appealing against its decision.
However, the council said planning reforms mean it has no chance of successfully challenging Ables appeal, so it will withdraw its objections.
In a statement on the councils website, council planning committee chairman Rob Cook said: The planning committee has met in recent weeks to review its position in the light of new Government guidance, which recognises the need for this country to develop facilities for ship dismantling.
"We have listened to advice from legal and other experts and felt we couldn't substantiate the reasons for refusal and therefore decided to offer no evidence to the appeals I firmly believe that the original decision in October to refuse the applications was the right decision at the time.
Despite the councils new position Able will still need to go through the appeal process. But chairman and chief executive Peter Stephenson hopes to have planning approval by Christmas.
Stephenson said that the Government had taken an interest in the case and he suspected that this is because it is aware of the desperate need in the UK for ship processing plants.
He added: We have been horrified at the time it has taken to obtain permission to recycle ships at Teesside Environmental Reclamation and Recycling Centre (TERRC).
"We originally applied for planning permission in 1996 and planning officers had recommended the plans for approval. We think that decisions should be based on technical merit, and in this case we think it was there.
We still want TERRC to be a centre of ship recycling excellence.