Michelle Kelly and one-year-old Jaden, from Mellow Purgess, Basildon, Essex, were reported missing last Friday and it is believed they could be in Scotland.
An Essex Police spokesman said: "Jaden is subject of a care order with parental responsibility having been given to Essex Social Services.
"Following an unsuccessful appeal Jaden should have been in the care of Essex Social Services on the Friday. When social services and police attended her address, mum and son were not there.
"Officers are concerned for their welfare and are continuing to search for them.
"There has been information to suggest they may have gone to Scotland and inquiries are also being focused there."
Ms Kelly is 28, white, 5ft 4in and of heavy build. She has shoulder-length blonde hair which may have been dyed a plum colour. She speaks with a London accent.
Jaden is described as being of "mixed heritage" with curly dark brown hair.
Superintendent Nick Morris said: "I would ask anyone who sees a mother and young child matching these descriptions, who they have not seen in their area before, to contact us.
"We are concerned for the welfare of both mother and child and need to find them as soon as possible."
Speaking directly to Ms Kelly, he added: "We know you love Jaden and would not see him come to any harm.
"I know how difficult this must be for you; however, we need to make sure you are both OK. Please call police or your mum and let us know where you are."
Anyone with information about Ms Kelly and Jaden's whereabouts is asked to contact police on 101.