The baby boy was found on The Mounds, off Shannon Road, in Kings Norton at around 5.40pm.
Police believe he was just hours old when he was discovered by a dog walker, clothed and wrapped in a blanket.
He was found the day before St. George’s Day, so has been named George by the hospital staff looking after him.
Police say baby George is healthy and doing well, but they have still been unable to find his mother.
Officers say they are very concerned for his mother and want to check she is doing OK, both physically and mentally.
A spokesman for West Midlands Police added: "This is our only concern right now, giving birth is a big thing and we’re concerned she will need medical help."
Detective Inspector Neil Hunt, who is leading the investigation, said: “I am reaching out to George’s mother.
"You are my primary concern right now, and we really need to make sure you’re OK.
"I don’t know the circumstances around George’s birth but, as a parent myself, I know exactly how overwhelming becoming a parent can be.
“We’re ready to offer you any help and support you may need. George is safe and well, and your health and wellbeing is our priority - please be assured that you are not in any trouble.
“If anyone thinks they know who George’s mother is, I’d urge you to come forward to help me locate her − you can contact us in confidence.
“I’d also like to hear from anyone who was in the area between 5pm and 6pm last Thursday, April 22. If you noticed anything or saw someone, please contact us as a matter of urgency."
CCTV released over the weekend shows a woman walking along Shannon Road just before 5.10pm that afternoon − we’re still appealing to trace the woman in the clip as we believe she may have vital information about what happened.
When found, baby George was wearing grey leggings, an orange and white striped T-shirt with a dinosaur motif, and was wrapped in a grey and white blanket (see picture above).
He remains in hospital in the West Midlands.
Detectives are in the area today handing out leaflets, encouraging any witnesses or anyone who knows anything to contact them.
Anyone with information is urged to contact police via live chat at www.west-midlands.police.uk 8am − midnight, or by calling 101 anytime.
To remain 100 per cent anonymous, contact Crimestoppers on 0800 555 111. Quote crime number 3476 of April 22.