My understanding here is that you are looking for a job, getting interviews, and waiting for a formal offer.
You want to be able to jump ship as fast as possible, and for that you want to remain available and avoid getting into new projects/tasks that would prevent you from asking for a shorter notice period.
- Your next employer is aware of the 3-month notice. They also know that you can ask to shorten it. They also know that your current employer does not have to accept a shorter notice.
- They will ask you to shorten your notice, and possibly make it part of the negociation.
- Your current employer will want you to complete your tasks but they won't want you to stay if you have nothing else to do and/or are not motivated.
When you are given new tasks, make sure to prepare your exit:
- Document what you are doing (more than usual)
- Work with other people whenever possible
- Delegate if you can
- Cut tasks in smaller chunks that you will be able to complete rapidly, so you can pass the next chunks to someone else more easily
- All this will help you evaluate how much time you need to pass you work to next person/reach a point in your work where it makes sense to stop
From experience, it is a negociation between
- you and your future employer to get as much slack as possible regarding the starting date. You can mention that you want to take some time of, that you want to finish your project wherever you are now, that your current employer won't let you go right away
- you and your current employer. Show good will, demonstrate when you will be able to complete your tasks/pass the relay to someone else, and agree on a middle ground. They'll want you to stay as long as possible with motivation, not chain you to your desk. Paying you 3 month for half-assed work vs 6 weeks of proper transition.
Also your manager can be a good person. If you have a good relation with them, you can be direct: "I have this cool opportunity, but they need me to start in 6 weeks. I'd appreciate it if you could agree to shorten the notice period."
But you can't mention anything to your current employer until you either have a firm offer, or have decided when you want to leave no matter what (in which case, ponder what you'll do if you are out of a job at this point. Make sure you have a backup plan. Hand you resignation 3 month before). If you do, you are stuck to either accept whatever offer you finally get (if any at all), or go unemployed for a period of time. Just because interviews went well, don't assume you'll get an offer. Just because they verbally agreed to employ you doesn't mean it is true, nor that it will be next month.
The only case where you'll want to indicate that you are either looking for another job or responding to offers is when you want to give a last chance to your manager to change something (project, team, career orientation...) and have already discussed the discomfort of your current situation. It is a tool to handle with care, as it is the last one. Your manager can (depending on the laws in your country) let you go, or stop giving you project if you tell them that.
So, in short:
- Get a starting date as far as possible
- When you have a signed offer, resign
- Negociate the end date of your current contract
- Meanwhile, when given new tasks, prepare your exit