Carew is available as a free agent after leaving Aston Villa at the end of last season.
Allardyce told Sky Sports News: “There has been an interest in John. I have had a chat with him, but he has a number of options on the table.
“We’ve given him our terms and conditions and we will try, if we can, to go from there. But whether we can negotiate that or whether he will accept that is obviously another matter.
“He is another quality player who has played at most levels of football in his lifetime and he’s only just in his 30s, so he’s a very mature, experienced and good player.
“If he chooses us we would be delighted.”