The investigation into the child's death is being led by Detective Inspector Euan Morrison of Lothian and Borders Police.
The naked body of the baby boy - which was said to be charred - was discovered on Sunday morning on a footpath near Harewood Crescent in Edinburgh's Craigmillar area.
A police spokesman said that a number of people had contacted them and they were following several definite lines of inquiry.
A post mortem examination has been carried out on the infant's body but police have made no further comment.
The investigation into the child's death is being led by Detective Inspector Euan Morrison of Lothian and Borders Police.
He confirmed on Monday that the death was being treated as suspicious and he urged anyone with information to contact police immediately.
Mr Morrison said his officers were particularly keen to speak to the mother of the dead baby.
"We are appealing for witnesses in relation to the discovery of an infant child who was found yesterday morning at about 0800 GMT," he said.
"We are appealing for anyone who may know the youngster's mother to come forward or indeed, for the mother herself to get in touch with us.
Appeal for information
"A gentleman out walking his dog found the body and we are appealing for anyone who may have been in the area at the time or in the 24 hours before to come forward."
The Inspector said the child's body was discovered on a footpath next to "busy" playing fields.
He said a football match, attended by around 60 people, had been taking place on the playing fields on Saturday morning.
Mr Morrison said police did not yet know how long the baby had been lying before it was discovered or how long it had been dead.
He also refused to speculate on the age of the child or on reports that the corpse had been burned.
But he did confirm that the body was naked when it was found and that it showed signs of injury.
"The body was lying beside the footpath and was clearly visible to the gentleman who discovered it," he said.
"There were some injuries to the body but I can't go into any specific details."
The Inspector said that a number of local people had come forward with information and his officers were following several lines of inquiry.
Shocked community
He added that local community health workers had also spoken to police and he said they would be trying to locate women who have recently given birth in a bid to shed more light on the mystery.
Local people reacted with shock to the news that an infant's body was found on the estate and some left flowers near the scene.
One woman said: "Everyone in the street just can't believe something as awful as this could happen.
"The path is used by a lot of people and is very busy on Saturday nights."
Cameras trained on the path may have caught what happened.
Anyone who may have information on the incident is asked to call Craigmillar police station on 0131 652 4133.