The Norwegian frontman is available as a free agent after severing ties with Aston Villa.
He was deemed surplus to requirements by Gerard Houllier last season and spent the second half of the campaign on loan at Stoke.
Various clubs have been linked with a move for his services over the summer, but Allardyce has revealed that discussions have been held with the 31-year-old and that terms have been put to him.
He is now waiting to hear back from Carew and believes adding another proven top-flight performer to their ranks would only serve to aid the Hammers' promotion push in 2011/12.
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."