Samantha Baldwin is wanted on suspicion of abducting Dylan Madge, six, and Louis Madge, nine, for whom she has no legal custody at this time.
She was last seen in Nottingham City Centre at around midday on March 27, and South Yorkshire Police have this afternoon asked for people in the region to help assist Nottinghamshire officers in their investigation.
Police believe she is with Dylan and Louis.
Officers have made two arrests in relation to the disappearance of the mother and two boys.
A 62-year-old woman and a 36-year-old woman were arrested on suspicion of assisting an offender and remain in police custody.
Superintendent Rich Fretwell said "An investigation into the circumstances is well underway and we have a large amount of resources working around the clock to locate Dylan and Louis. Our priority at the moment is to locate them safe and well.
"We are now appealing for the return of Dylan and Louis and any information about the whereabouts of them and their mother, Samantha."
Samantha is described as being 5ft 7ins tall, of a slim build, with shoulder-length, highlighted blonde hair and green/blue eyes. She was last seen wearing a red knee-length coat from Mango but may have changed her clothing since.
Louis is described as having collar-length straight fair hair, a fair complexion and blue eyes.
Dylan is described as having collar-length straight dark brown hair, a tanned complexion and brown
Officers do not have a description of the clothing the boys were last seen wearing.
In the last 24 hours, we have seen thousands of messages of support and we have been overwhelmed by members of the public offering their help. We do not underestimate just how important information from the public is and we would like to say thank you for this continued support.
We ask anyone with information of their whereabouts to call Nottinghamshire Police on 101 urgently, quoting incident number 332 of 27 March 2017.