Bryson, a serial sex offender who attacked vulnerable women when she was known as Adam Graham, decided to transition while awaiting trial, which provoked outrage among campaigners.
It is now understood prison chiefs have in recent weeks approved the move of Tiffany Scott, who stalked a 13-year-old girl while known as Andrew Burns.
It is believed Scott will switch from a male wing to an all-female unit.
It is unclear where this location will be.
Cornton Vale is Scotland’s only all-women site, but female prisoners are also detained at other jails.
The Scottish Conservatives have told Sky News they want Ms Sturgeon to again block the move.
‘Absolutely appalling’
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The party’s equalities spokesperson, Rachael Hamilton MSP, said: “The fact that such a violent and dangerous criminal is set to be transferred to a women’s prison is absolutely appalling.
“It is clear that Tiffany Scott continues to present a grave risk to the safety of any women that come in contact with them – even trained prison staff.
“The idea that this violent individual may soon have access to scores of vulnerable women within our prison estate is truly repellent.
“Women’s safety must not take a backseat to the wishes of violent criminals.
“Nicola Sturgeon has already U-turned under public pressure to belatedly remove double rapist Isla Bryson from a women’s jail, she must now intervene to block the transfer of this violent individual before it takes place, or knowingly risk the safety of some of Scotland’s most vulnerable women.”
The Scottish Prison Service (SPS) said it does not comment on individual cases, but confirmed it was drawing up fresh proposals on dealing with trans prisoners.
An SPS spokesman said: “We do 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.
“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.”