Webster, 54, was jailed for a minimum of 30 years in 2011 for murdering his first wife Claire Morris in a faked car accident in 1994.
He was also convicted of trying to kill his second wife, Aucklander Felicity Drumm, in a similarly staged car crash in New Zealand five years later.
BBC Scotland reported The Scottish Criminal Cases Review Commission (SCCRC) will compile a report of the case over the coming months.
The SCCRC will then decide if it feels the case should be referred back to court.
Webster has previously lost an appeal against his conviction.
Webster, a former nurse from Surrey in England, stood to become a millionaire thanks to the life insurance payouts from his first wife's death.
It was only when Drumm survived her crash and Webster disappeared with her life savings that prosecutors re-examined the earlier death.
A 2011 trial heard how Webster drugged his first wife Ms Morris just eight months after their marriage and drove his car off the road with the unconscious woman inside.
He then torched the vehicle and covered his tracks before collecting the life insurance payout and moving to New Zealand.
The jury then heard Webster married Ms Drumm before attempting a similar deadly scam in 1999 near Auckland by drugging her and planning to kill her in another staged smash.
On his return to Scotland, he tried to con another woman into a bigamous marriage, and even getting her to change her will to leave him everything, including her house and a luxury yacht.
By NZME. News Service.
