The giant striker feels settled at Villa and believes his style of play is suited to the Premier League.
The 28-year-old joined Villa from Lyon in a swap with Milan Baros in January 2007 and was their leading scorer last season with 13 goals.
He has two years remaining of his current deal but would be happy to extend that if Villa are of the same opinion.
Carew said: "I have two years more on my contract so there is no rush with anything. But I am very happy to be here and I would like to stay for many years more.
"If that is the club's wish, then that is my wish as well. We will see what happens. But I would like to stay here for a long time.
"I really like it here, am really happy here, so I am absolutely open for that. I would like to put roots down here for a quite a long time.
"It is a league that really suits me well and I am happy with the football and really enjoy playing it. That is important to me."