Handing several months notice at work

Hi, I am soon to be handing in my notice at work. I have received a job offer in writing pending successful background checks and references. One reference has already come back excellent but the other is in the hands of my current employer who I need to inform of my notice. Is giving my current employer several months notice too early? It’s just an awkward situation as my new job want to contact my current employer but out of respect I want to speak with my boss first.

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