It is understood that Tiffany Scott, who was previously known as a man called Andrew Burns, has applied for the move.
The Daily Record first reported that the transfer has been approved and is likely to happen later this year.
The Scottish Conservatives have called for the move to be stopped.
A spokesperson for the Scottish government said it was an operational matter for the Scottish Prison Service.
The prison service said decisions about where to place transgender prisoners were made on "an individualised basis".
It follows the imprisonment of Isla Bryson - a transgender woman convicted of two rapes before she changed gender - in Cornton Vale women's prison.
She was later moved to a male wing of HMP Edinburgh.
Scott is serving an indefinite sentence as she is subject to an order for lifelong restriction (OLR), meaning she will only be released when she is no longer considered an "unmanageable risk to public safety".
She was handed that sentence in 2013, after admitting stalking a 13-year-old girl by sending letters from her cell.
In 2017 Falkirk Sheriff Court was locked down amid safety fears as Scott was sentenced over a series of violent incidents in Glenochil Prison in Clackmannanshire.
They included striking a prison nurse on her back with a hurled chair, punching a prison officer in the face, and spitting at another officer and trying to bite him.
She also smeared excrement over her cell, tore a drip needle out of her arm, and ripped up "tear-proof" clothing.
The Scottish Prison Service said it would not comment on individuals.
"Decisions by the SPS as to the most appropriate location to accommodate transgender people are made on an individualised basis, informed by a multi-disciplinary assessment of both risk and need," a spokesperson said.
"Such decisions seek to protect both the wellbeing and rights of the individual as well as the welfare and rights of others around them, including staff, in order to achieve an outcome that balances risks and promotes the safety of all.
"Where there are any concerns about any risks posed by an individual, either to themselves or others, we retain the ability to keep them separate from the mainstream population until an agreed management plan is in place."
A Scottish government spokesperson said: "The placement of prisoners is an operational matter for the Scottish Prison Service who use comprehensive individualised risk assessments to inform decisions, such as the appropriate location of transgender people in custody.
"SPS is reviewing their policy on managing transgender prisoners in partnership with the Scottish government and that process is nearing completion."