City went back to a circle crest after years of not using one, but the designers of the NYCFC crest don't appear to have been inspired by the Man City's old crest that they weren't using. NYCFC's crest came out a year and a half before Man City updated theirs back to a circle.
NEW YORK CITY FC - Matthew Wolff
Although I didn’t realize it at the time, the crest I created for NYCFC would massively influence the crest design for sister clubs Melbourne City FC and Manchester City FC.
Other designer Rafael Esquer.
Interview with Rafael Esquer, History Behind NYCFC Badge | The NYCFC Nation
He didn’t just stop looking at New York City for inspiration. He wanted to design a badge that was timeless and could fit in anywhere in the world, from MLS to England or Australia.
Funny enough, City Football Group loved his design ideas so much that they took the idea from NYCFC’s logo and ended up incorporating it into both Melbourne City and Manchester City’s badges, giving them new looks based off of NYCFC.
The style of all 3 crest are very similar in how they are laid out. Multiple circle borders around the outer crest on all 3. NYC has the pentagon, Melbourne the hearts and Man City the 18 and 94 all placed in the exact same places on crest. Thick circle border on the inside of the crest that borders the teams logo.
So yes, NYCFC's style of circle crest did inspire Man City when the decided to go back to a circle crest.