Wilshere was in charge of Norwich for their final two Championship games of the season following the sacking of Johannes Hoff Thorup. He led his side to a 0-0 away draw against Middlesbrough and a 4-2 home victory over Cardiff City on the final day of the season but the results were not enough to impress the board.
Canaries sporting director Ben Knapper told BBC Radio Norfolk that the club's search for a permanent replacement will be "focused externally," which means Wilshere is not a candidate for the post.
"Jack is a fantastic person, he and I have a big history together," said Knapper, who worked at Arsenal when Wilshere was a player there.
"He's a great coach and will be a fantastic head coach, but that's not going to be here at this particular moment in time.
"Naturally, he's disappointed. He's super-ambitious, and he should be, but for where we are right now and the context, we're focusing externally.
"The emotional part, the romantic part of me would love to make that decision but my job is to not get wrapped up in emotion and to make cool, objective decisions."
Wilshere had been working at Arsenal as their Under-18 manager before his spell with the Championship side, where he led them to the final of the FA Youth Cup in 2023.
His future remains uncertain as he looks for a new adventure in management ahead of the new campaign, but what is certain is that many clubs will be asking about his availability after an impressive start to coaching.