The giant Norway international was reported to be attracting interest from both Manchester City and Tottenham.
Villa have already seen midfielder Gareth Barry depart for Eastlands and O'Neill does not want to lose the services of another key man.
Carew ended the season as Villa's top scorer with 15 goals, despite spending a large chunk of the campaign on the sidelines with a back injury.
O'Neill told the Daily Star Sunday: "We want John to stay at the club - he's signed for a couple of years and we and we want him to stick around.
"He's very popular with the crowd and that helps his confidence. And his goalscoring ratio is now up where it should be.
"We want him to stay here, it's a simple as that."