The midfielder has been linked with a move away from Liverpool this summer.
Liverpool's Joe Allen warms up before the match
The Welshman has been heavily linked with a departure from Anfield having fallen down the pecking order under Jurgen Klopp.
While he featured under the German last season and made 19 appearances, most of those were from the bench as he managed just 731 minutes of action.
Liverpool's Joe Allen
That has led to talk he will leave the club as he looks for regular first team football to stop his career stagnating on Merseyside.
He is well liked at the club though, and Carragher has said it would be unfortunate to see him depart.
“He has been at Liverpool for three or four years, and he has done well, but it probably hasn’t quite gone as according to plan as he would have liked,” he told the Liverpool Echo.
“It will be unfortunate if he moves on, but I think Jurgen Klopp is looking for a player in that position, so if a good offer came in for him then fair enough, because they certainly shouldn’t sell him for buttons.”
Liverpool's Joe Allen in action with Villarreal's Jaume Costa
It now remains to be seen whether Allen will depart Liverpool this summer after his performances at Euro 2016.
In a summer transfer window of high transfer fees, his impressive displays are likely to have pushed his asking price up further.
With the Reds keen to already recover the £15 million they originally paid for Allen, it will be interesting to see if anybody is willing to pay the fee they will ask for him.
Liverpool's Joe Allen in action