The Alan Partridge star was seen with several members of the cast and crew on location in Morningside on Monday, May 10.
The film will tell the true story of how the remains of King Richard III were discovered buried under a carpark in Leicester in 2012.
Oscar-nominated Coogan plays the husband of historian Philippa Langley, who had a key role in unlocking the mystery behind the Plantagenet king’s final resting place.
On Monday, bystanders in the leafy suburb watched as Mr Coogan rehearsed a scene with director Stephen Frears (‘The Queen’, ‘Dirty Pretty Things’) alongside the two child actors who play his sons.
Footage from the rehearsal shows the two boys, dressed in school uniforms, rushing from a garden path towards a parked car followed by Mr Coogan, who ushers them into it.
But as the 55-year-old makes for the driver’s seat and utters his line, the director intervenes.
“Say the line before getting in,” Mr Frears can be heard shouting, before the two BAFTA-winners discuss the scene together in more detail.
‘The Lost King’ is the latest artistic collaboration between Mr Coogan and Mr Frears, who achieved widespread critical acclaim with ‘Philomena’ in 2013.
Meanwhile, Sally Hawkins (‘The Shape of Water’) will play Ms Langley in ‘The Lost King’, who put her remarkable discovery of Richard III’s remains down to gut instinct.
It is understood that once principal photography has concluded in Morningside, the production will recreate the famous Leicester carpark excavation at the National Mining Museum’s Lady Victoria Colliery in Newtongrange.