Local reports said Peshkov will be buried on Wednesday after his body arrives in his hometown Lipetsk, an industrial city in central Russia.
Dmitry Peskov, the spokesman, told reporters that Mr Obama expressed regret over the incident and discussed the need for a political solution to the Syrian crisis.
Mr Erdogan called for face-to-face talks with Mr Putin last week as the two countries traded blows over who was responsible for the incident.