The brief, to provide pan-European strategy and co-ordination of the agency network in specific markets, is thought to be worth a low-end six-figure sum.
Cake will cease to work for the world's number one gaming company in October, but emphasised that the split was amicable.
Fishburn Hedges chairman Neil Hedges declined to comment.
Weber Shandwick is understood to be one firm that made it through to the final round of the pitch process, which was managed by AAR.
Nintendo pitched out the brief because its business needs have changed from Cake's purely consumer PR approach to a more full-service requirement, with specific expertise in public affairs and corporate reputation now seen as essential.
Nintendo head of PR Shelly Pearce said: 'The time has come to adopt a different approach to our business needs which means we require a different kind of support.'
Cake Group CEO Mike Mathieson said: 'We had an honest and open discussion about it with Nintendo. For some months, the business has seemed to be more about protection than outreach. We are not good at that.'
Mathieson added that his agency would now be actively seeking clients in the gaming sector and would continue to be on Nintendo's roster for event-based activities.
Cake lost Nintendo's UK business to Red Consultancy last September. Red Consultancy will continue to provide Nintendo with support in the UK.