He was found in a bag wrapped in a single towel with his umbilical cord still attached.
Attracted by his cries, residents discovered him in the garden of a house in Liverpool.
They alerted police and the baby was taken to hospital.
Despite his unhappy start in life, Zack who weighs 8lb 10oz, has been described as a "picture of health" by medical staff.
Now they are now appealing for his mother to come forward.
Lydia Moore, head of midwifery at the Liverpool Women's Hospital, said: "Zack is doing very well.
"He is in good condition and is feeding well but we are concerned about Zack's mum.
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Baby Zack: found with his umbilical cord attached
"She has given birth on her own with no medical assistance. She needs to get some attention from a doctor or a midwife.
"We have no idea what kind of delivery it was. We would love it if she came and saw us here."
Nurses at the hospital named him Zack, after the Fazakerley district of the city where he was found on Thursday night.
Neonatal nurse Zishan Ahmet, who is one of the the team looking after Zack, said: "He is a lovely baby. He slept a lot and eats well. He had a bath this morning and he loved that.
"He is a good healthy weight and he loves cuddles.
"All the nurses are looking after him. Everyone keeps popping in and makes sure he gets lots of hugs. We have really been making a fuss."
Merseyside Police are appealing for his mother to come forward and have promised her "support and understanding".
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The garden in the Fazakerley area of Liverpool where the boy was found
Superintendent Colin Leeman said: "Zack is doing really well. Obviously we are also concerned for the well-being of his mother. She may be bleeding or may have suffered an infection."
Police are studying CCTV footage of the area where Zack was found and carrying out house-to-house inquiries in an attempt to trace his mother.
Residents who discovered him said his cries sounded like those of a distressed cat.
Mr Leeman added: "He was just an hour or hours old. Clearly this little boy needs his mum."
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