Wilshere, 18, joined Bolton on loan in the January transfer window and has impressed for the Trotters, making seven league starts and scoring his first Premier League goal in a 2-1 win at West Ham.
Impressing: Jack Wilshere tussles with Antonio Valencia during Bolton's clash with Manchester United on Saturday
The midfielder is the youngest player to represent Arsenal in the league and remains a part of Wenger's long-term plans at Emirates Stadium, but Coyle is hoping Wilshere will continue his education at Bolton next season.
'That's a conversation for the summer, but I felt he was terrific and showed a mature head on young shoulders,' Coyle told the Daily Star.
'That's great for Arsenal, Bolton at the moment, and English football in the future. If Arsene Wenger wants to give him to me for another year, I'd be delighted to take him.'
Bolton boss Owen Coyle slams Manchester United's Nemanja VidicI'm ready, Fab: Bolton ace Muamba dreams of senior England call-upBOLTON WANDERERS FC