The 22-year-old was imprisoned for cocaine smuggling in 2013 and had appealed to be expelled to serve the remainder of her sentence closer to home.
Reid, from Lenzie, East Dunbartonshire, and Michaella McCollum, from Dungannon, Co Tyrone, were jailed in 2013 for six years and eight months after admitting trying to smuggle cocaine worth £1.5 million from Peru to Spain.
McCollum, 23, was freed in March under new legislation on early prison release introduced in Peru last year after serving two years and three months.
According to the BBC, Peruvian authorities said they had agreed to release Reid after a court hearing on Friday.
A magistrate ruled that she had been remorseful and shown her “intention to be re-inserted back in society’ at an earlier hearing.
The Daily Mail reported that the court was to inform the British embassy of the decision so arrangements can be made for Reid to be returned to the UK.
The Foreign Office said on Saturday it was providing assistance to Reid and remained in contact with her family and local authorities.
However it is anticipated to be some time before the women are able to return home.
McCollum and Reid were caught with the haul at Lima airport on August 6 2013 while attempting to fly to Spain.
They had claimed they were forced into carrying the drugs but pleaded guilty to charges later that year.
The pair were caught trying to board a flight with 24lb (11kg) of cocaine in food packets hidden inside their luggage.
McCollum and Reid faced the prospect of a maximum 15-year prison term but struck a behind-closed-doors plea bargain to secure a shorter sentence.
They had previously been held at Lima’s Virgen de Fatima prison but were moved to the Ancon 2 prison, where McCollum was reportedly crammed into a cell with 30 other prisoners with poor sanitation and toilet facilities.
The SPS agreed in principle to a transfer in 2014 and Reid has been awaiting approval from the Peruvian authorities, who need to consent to her serving the remainder of her sentence under Scots law.
Reid’s father Billy has previously said the impact of his daughter’s crime on his family had been “horrendous” and spoke out in a video warning of the consequences of drug offences abroad.
He said: “It’s horrendous to see your daughter in handcuffs and the living conditions that she has to put up with. Melissa has spent her own 20th and 21st birthdays in prison in Peru.
“She missed the significant event of her only brother’s wedding. Events such as Christmas are non-existent for us.
“There’ll be no celebrations in our house, there’ll be no Christmas tree until we get her back home.’’