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As per Verse 1.59 and 1.61, the Manu of Manu Smriti is the first Manu named Svāyambhuva Manu of first (Svāyambhuva manvantra):

[Manu:]

This Bhṛgu will fully describe to you this (Law); this sage has learnt the whole of this in its entirety, from me.—(59)

[Bhṛgu:]

स्वायम्भुवस्यास्य मनोः षड्वंश्या मनवोऽपरे ।

सृष्टवन्तः प्रजाः स्वाः स्वा महात्मानो महौजसः ॥ ६१ ॥

There are Six other Manus, high-souled and mighty, who belong to the same race as this Svāyambhuva Manu, and have called into being, each his own offsprings.—(61)