Were it that I was coaching you, I'd recommend you start by gradually ramping your mileage up to into the 40-50 miles per week range, and once there, do that for at least six months. With at least a thousand miles of nice easy paced running under your belt, start working speed drills into the mix, still keeping your mileage up to at least 40 miles per week. By this time next year you will have significantly lowered your 10K time and you should have a pretty good idea if a 10 minute 3K will be even remotely possible.
Keep in mind, there are many people in this world that no matter what they do for training will never be able to sustain a 5:20 per mile pace for a single mile much less for 1.86 miles (a 3K).