This is a second answer I will provide for this question from an article on Anapanasati I recently published.
The preliminary instruction in the Anapanasati Sutta is the following stock phrase found in many suttas:
Here a bhikkhu, gone to the forest or to the root of a tree or to an
empty hut, sits down; having folded his legs crosswise, set his body
erect, and established mindfulness to the fore (parimukhaṁ), ever
mindful he breathes in, ever mindful he breathes out.
While the above included term ‘parimuka’ is often the subject of debate, the preliminary instruction here in MN 118 in no place refers to intentionally directing mindfulness onto the breathing. As previously suggested, directing mindfulness onto the breathing appears not possible because mindfulness does not mean consciousness, observing, contemplation, attention or similar terms.
In tune with the meaning of ‘mindfulness’ in SN 35.245 as ‘the gatekeeper’, MN 38, while not mentioning Anapanasati but instead proceeding directly to jhana, provides the following instructional explanation of the above stock phrase:
Endowed with this noble aggregate of virtue, this noble restraint over
the sense faculties, this noble mindfulness & alertness, he seeks out
a secluded dwelling: a wilderness, the shade of a tree, a mountain, a
glen, a hillside cave, a charnel ground, a forest grove, the open air,
a heap of straw. After his meal, returning from his alms round, he
sits down, crosses his legs, holds his body erect, and brings
mindfulness to the fore.
Abandoning covetousness with regard to the world, he dwells with an
awareness devoid of covetousness. He cleanses his mind of
covetousness. Abandoning ill will & anger, he dwells with an awareness
devoid of ill will, sympathetic with the welfare of all living beings.
He cleanses his mind of ill will & anger. Abandoning sloth &
drowsiness, he dwells with an awareness devoid of sloth & drowsiness,
mindful, alert, percipient of light. He cleanses his mind of sloth &
drowsiness. Abandoning restlessness & anxiety, he dwells undisturbed,
his mind inwardly stilled. He cleanses his mind of restlessness &
anxiety. Abandoning uncertainty, he dwells having crossed over
uncertainty, with no perplexity with regard to skillful mental
qualities. He cleanses his mind of uncertainty.
MN 38 (Thanissaro Bhikkhu translation)
Similar to the MN 38 instruction above, when the word ‘mindfulness’ is actually used in a practical sense in MN 118, following the stock definition of Samma Sati found in many suttas, mindfulness is used in the
….on that occasion a bhikkhu abides…mindful, having abandoned longing
and dejection in relation to the world.
It follows when MN 118 describes mindfulness as a factor of enlightenment, it appears to say much ‘noble’ mindfulness possesses/matures with the salient quality of ‘letting go’, ‘surrender’ or ‘relinquishment’ (namely,
‘vossagga’), as follows:
Here, bhikkhus, a bhikkhu develops the mindfulness enlightenment
factor, which depends on seclusion, dispassion and cessation that
matures as relinquishment/abandonment/surrender/letting go (vossagga).
Similarly, in respect to the development of concentration (samadhi), SN 48.9 & SN 48.10 unambiguously say concentration & jhana are developed by making ‘vossagga’ the object of the mind (aramana), as follows:
And what is the faculty of concentration (samādhi)? It is when a noble
disciple, making letting go the object (vossaggārammaṇaṁ), gains
concentration, gains unification of mind. This is called the faculty
SN 48.9 & SN 48.10
Also, as previously mentioned, MN 117 & SN 46.3 say the role of mindfulness is to bring the Teachings to mind and remaining abiding in Right View. Naturally, ‘vossagga’ is a manifestation of Right View, namely, per the instruction about the 2nd Noble Truth in SN 56.11, the abandonment of craving. Thus the 3rd Noble Truth contains a Pali word similar to ‘vossagga’, having the same root, namely, ‘paṭinissagga’.
In conclusion, when the opening instruction say the practitioner “brings mindfulness to the fore” (parimukhaṁ) and is “ever mindful”, it appears this means to always maintain Right View (per MN 117), the wholesome (per MN 38),
the absence of longing & dejection (per MN 118) and/or vossagga (per MN 118, SN 48.9 & 10) in the mind.
To reiterate, in no place is an instruction to direct attention/consciousness towards/onto the breathing. In other words, since the nature of consciousness is to come into contact with the most salient sense object, a mind established in letting go (vossagga) will naturally be quiet and the in & out breathing will automatically become the object of consciousness. By being free from coarse willfulness to observe the in & out breathing, the experience or perception of the in & out breathing will be much more lucid, pervasive & long-lasting.