Reid, aged 20, from Lenzie, East Dunbartonshire, and McCollum, aged 21, from Dungannon, Co Tyrone, were jailed last year for six years and eight months.
They admitted trying to smuggle cocaine worth £1.5 million from Peru to Spain.
A solicitor has described conditions in detention in the South American country as horrendous. Both have now been successful in their bids for repatriation, with the Scot likely to return to a Scottish prison. A prison service spokesman confirmed that it has now completed the paperwork relating to a transfer, and is awaiting the next move by the Peruvian authorities.
It is likely that Reid will serve out her sentence at Cornton Vale prison, Scotland's only women's prison, which is near Stirling. It is only about 25 miles from her family's home.
The Irish government has been providing consular support to McCollum, an Irish citizen.
A letter from the Republic's department of foreign affairs said: "The Peruvian authorities have confirmed that they have accepted Michaella's prison transfer request and have passed this to the UK National Offenders' Management Service (NOMS) which co-ordinates prisoner transfers to the UK.
"In Ms McCollum's case, NOMS liaises with the Northern Ireland Prison Service and with the Peruvian Prison Service on all aspects of the transfer."
It could be months before she returns home as the logistics of the transfer will be complicated, the note to McCollum's solicitor stated.
Prisoners must be accompanied throughout their journey; airlines and airports must be advised, with security arrangements put in place at departure, transit and final stops.
A Northern Ireland Prison Service spokesman said: "We don't normally comment on individual cases. All transfer requests are however dealt with as expeditiously as possible."
The pair were caught with the haul at Lima airport on August 6 last year.
They were working on the Spanish party island of Ibiza when they claimed they were forced by Colombian drug lords who kidnapped them at gunpoint to board a flight with 24lb 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 notorious Ancon 2 prison, where horrific conditions reportedly mean McCollum is crammed into a cell with 30 other prisoners.
The situation at the mixed prison, which is two-and-a-half hours from Lima, has previously been criticised by the Irishwoman's lawyer as "appalling".
Kevin Winters said sanitation and toilet facilities are extremely poor and all females have to use a hole in the ground which has to be covered up because of the presence of vermin.
Reid's father, from Lenzie, near Glasgow, has said he has met Scotland's justice secretary Kenny MacAskill and is hopeful that his daughter will serve the rest of her sentence closer to home.
Patrick Corrigan, Amnesty International's Northern Ireland director, said: "With massive overcrowding, poor access to medical care and endemic corruption throughout Peru's penal system, it is understandable that prisoners from the UK and Ireland, such as Michaella McCollum Connolly, want to return home to complete their sentences."
He said Peru's prisons are full to overflowing, with almost 50,000 prisoners for fewer than 30,000 prison places.
"The Virgen de Fatima and Ancon jails, where Melissa Reid and Michaella McCollum Connolly have been serving their jail terms, suffer from the same problems as the rest of Peru's prison system, which has a reputation for being deeply corrupt, with prisoners even forced to pay for food and clean water.
"Significant investment is required to bring the country's jails up to internationally accepted standards for prison conditions such as would be expected in the UK."
The Scottish Prison Service (SPS) said it has approved a repatriation application it received from Lima earlier this year.
Prison chiefs are now waiting for the Peruvian authorities to agree on the terms of the transfer before it can take place.
An SPS spokeswoman said: "We have provisionally given consent to the Peruvian authorities for the application."