Bra entrepreneur Michelle Mone initially phoned police after allegedly spotting a baby in a vehicle without its parents.
Ms Mone was Christmas shopping on December 22 when she said she noticed the baby in the car park of Costco in Springburn.
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She phoned police and officers arrived to investigate, along with paramedics, to assess whether the child had been harmed.
It is understood that Mulgrew, 24, and 22-year-old Alana have hired a legal team headed up by Paul McBride QC.
A Crown Office spokesman confirmed that a report had been made to the procurator fiscal.
He said: “The procurator fiscal at Glasgow has received a report concerning a 24-year-old male and a 22-year-old female in connection with an alleged incident on December 22, 2010.
“The report is under consideration.”
Strathclyde Police confirmed that a couple have been charged.
Police questioned the couple at the time of the incident and initially cautioned them before charging them under Section 12 of the Children and Young Persons Act.
The baby was examined by paramedics, but was found to be unharmed.
The pair have reportedly told friends the matter is “a big misunderstanding”.
The couple insist they had only gone in to the store briefly and kept a close eye on the car.
A source close to the couple said: “The pair of them are devastated but are preparing for the likelihood of a court case.
“They deny any wrongdoing whatsoever – they had the car in sight at all times and it was locked because the baby was sleeping.”
After the incident, Ms Mone used online social network Twitter to give her account of why she had called police.
“Myself ... and store staff were extremely concerned about a baby left in a car, not in sight of front door and car alarm going off,” she wrote.