Wilshere is available after leaving his coaching post at Carrow Road.
Plymouth, following relegation from the Championship, are seeking a new manager after Miron Muslic quit for Schalke last week.
Muslic's decision left the Argyle board furious, which they made clear when announcing his departure.
The club's statement read: "Argyle can confirm that we have reluctantly allowed Head Coach Miron Muslic to open talks with a 2. Bundesliga side after he expressed his desire to speak with the German team and not return to England.
"Despite regular communication since the end of the season, and no mention of any issues, Miron has now made it clear that, after extended periods away from his family, this opportunity provides him the chance to be closer to them after four years on the road."
Frustration
The announcement continued: "We are frustrated by this decision, having implemented clear structures, processes and personnel to aid Miron and his coaching staff as we look to return to the Sky Bet Championship at the earliest opportunity.
"Not only that, the club have also moved quickly to bring in new signings and received repeated verbal promises from Miron that he was committed to Argyle and would do everything to help the club bounce back to the second tier.
"Muslic joined the Greens in January 2025, on a three-and-a-half-year contract, after a period out of the game having departed Cercle Brugge.
"He will now leave Argyle just four months into that long-term commitment."
Wilshere took charge of the final two games of Norwich's season after the dismissal of Johannes Hoff Thorup.