The child is believed to be with James Dempsey - a man known to the child's mother - West Midlands Police said.
The 35-year-old is believed to be driving a silver Vauxhall Astra with the registration NH05 OWP.
The vehicle was last seen on the A45 heading towards Coventry just before 1am on Thursday.
The force is urging anyone who may have information to contact them on 0800 056 0944.
Detective Chief Inspector Ian Ingram said: "I am concerned for the safety of this baby and am appealing for help to find him.
"His family are anxious and worried about the baby boy and just want him home safe and well.
"I am appealing directly to James to contact us or the baby's family so we can arrange for him to be reunited with his mother - or to take the baby to a place of safety, such as a hospital.
"They could still be in the Birmingham area but it's also possible that they've travelled elsewhere in the country, which is why we are appealing to the public for help.
"I would ask everyone who is out and about this evening - wherever you are - to please keep your eyes peeled for Dempsey."
Mr Ingram urged members of the public to share the appeal on social media to help raise awareness.
To contact West Midlands Police with information call 0800 056 0944 or 999.