The 21-year-old, jailed for cocaine smuggling in 2013, has been seeking to serve the remainder of her six-year sentence closer to home in Scotland.
Reid, from Lenzie, East Dunbartonshire, and Michaella McCollum, from Co Tyrone, were caught with £1.5 million worth of drugs at Lima airport.
It is understood a judge has approved a prison transfer request and officers from the Scottish Prison Service (SPS) are ready to travel to Peru to collect Reid.
An SPS spokeswoman said: “We have not yet received any formal notification. When the application has been approved, the necessary measures will be put in place for the transfer.”
Reid and McCollum were working in Ibiza when they claim they were forced to board a flight with 24lb of cocaine in food packets hidden inside their luggage.
They were caught on August 6 2013 and later pleaded guilty to drug smuggling, each receiving a jail term of six years and eight months.
The SPS gave their agreement in principle to a transfer last year 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.
The repatriation of McCollum to Northern Ireland was approved by Stormont’s justice minister in January this year but, like Reid, she remains in jail in Lima for now.
The pair had previously been held at Virgen de Fatima prison but were later moved to the Ancon 2 jail, where conditions are said to be cramped, with poor sanitation and toilet facilities.
A spokesman for the Foreign and Commonwealth Office (FCO) said: “We continue to provide consular assistance. For reasons of consular confidentiality we cannot go into further details.”
Reid’s father Billy said recently that the impact of his daughter’s crime on his family had been “horrendous”.
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.”
Mr Reid spoke out in a video warning of the consequences of drug offences abroad.
More than 700 British nationals were arrested for drug-related crimes in 2013-2014, according to the Foreign Office.