Looking on the internet deeply has found these results:

engagement with is the most popular phrase on the web.

engagement by 1241,000 results on the web	engagement with 133,600,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
the act of ... Suggested citation. National Survey of Student Engagement. (2010). Major differences: Examining student engagement by field of study—annual results 2010. We more recently revised that resource as the PCORI Engagement Rubric, emphasizing the value of engagement by all healthcare stakeholders. The revised ... an engagement by contract involving financial obligation; "his business commitments took him to London". intervention, intercession commitment Looking Behind the Numbers: Practicing Engagement By Design. 4. Reflections: How Good Is Good Enough? The Role of Intentional, Inescapable Engagement.	I have a prior engagement with my boyfriend, sorry. But an engagement with Charles creates an alliance. There has been critical but constructive support for our engagement with Turkey in a very critical period. I have a previous engagement with a friend. And their individual visions, and their engagement with this mathematical mode. She called off the engagement with Charles for the sake of both our countries. All nicely commented on your engagement with the midwife. It was my engagement with you that ended that relationship.

engagement by

1241,000 results on the web

engagement with

133,600,000 results on the web

More popular!

the act of ...
Suggested citation. National Survey of Student Engagement. (2010). Major differences: Examining student engagement by field of study—annual results 2010.
We more recently revised that resource as the PCORI Engagement Rubric, emphasizing the value of engagement by all healthcare stakeholders. The revised ...
an engagement by contract involving financial obligation; "his business commitments took him to London". intervention, intercession
commitment
Looking Behind the Numbers: Practicing Engagement By Design. 4. Reflections: How Good Is Good Enough? The Role of Intentional, Inescapable Engagement.

I have a prior engagement with my boyfriend, sorry.
But an engagement with Charles creates an alliance.
There has been critical but constructive support for our engagement with Turkey in a very critical period.
I have a previous engagement with a friend.
And their individual visions, and their engagement with this mathematical mode.
She called off the engagement with Charles for the sake of both our countries.
All nicely commented on your engagement with the midwife.
It was my engagement with you that ended that relationship.