A police sergeant and constable are two of the officers served with misconduct notices by the Independent Office for Police Conduct (IOPC) over their handling of a stalking allegation made by Ms Spinks.
Ms Spinks, 23, was found fatally stabbed in a field in Duckmanton, Derbyshire on June 18, and is believed to have been killed by 35-year-old Michael Sellars.
She had reported Stellars to the police in February for stalking, several months before she was killed.
The IOPC also said three other Derbyshire Police officers could face misconduct proceedings over the steps they took after a bag of weapons, including a hammer, an axe and some knives, was discovered in May near where Ms Spinks was later found dead.
Ms Spinks was last seen alive by her mother at 7.30am on June 18, leaving the family home in Chesterfield to drive a few miles to Blue Lodge Farm stables in Duckmanton, where she kept her horse.
She was discovered unconscious at the stables half an hour later, fatally stabbed.
The body of Sellers, whom Ms Spinks’ family said was her former supervisor at a warehouse where she once worked, was found in a nearby field off Tom Lane a few hours later.
IOPC regional director Derrick Campbell said: "We have made good progress, but our investigation is ongoing and there is more work still to do."
Mr Campbell said: "After consideration of evidence so far, we have reached a stage where we have now served disciplinary notices on five officers.
"The purpose of such notices is to advise officers their conduct is subject to investigation, and do not necessarily mean that disciplinary proceedings will follow.
"The serving of notices will be kept under review. We aren’t able to provide more detail at present, but we appreciate the impact Gracie’s death has had on the local community and beyond and the importance of keeping people updated with our investigation’s progress.
"We are keeping Gracie’s family informed and we note the work they have undertaken with others to campaign for Gracie’s Law to help better protect victims of stalking."