The Tottenham defender, signed by Howe in September, has played a key role in helping Cherries maintain their promotion push.
Smith, 19, has missed just one league game since arriving on a youth loan arrangement, the agreement extended in January until the end of the season.
The Londoner has progressed through the ranks at White Hart Lane and his contract with Spurs is understood to be up for renewal in the summer.
Cherries have already held preliminary talks with Smith regarding the possibility of him signing a permanent deal at Dean Court.
However, they are likely to face stiff competition, with Howe's new club Burnley also believed to have sounded him out. Championship clubs Reading and Ipswich are also reported to be tracking him.
Cherries chairman Eddie Mitchell told the Daily Echo: We would love to get Adam signed up and are working hard to do that. He figures in our future plans and we would be disappointed to lose him. He is a great lad and has done fantastically for us since he has been here.
At this moment in time, we would love him to be part of our team for next season, whichever division we are in. We have put a proposition to Adam and are hoping he will take it up.
Smith, who has made 31 league and cup appearances for Cherries, is probably likely to keep his options open until the end of the season.