Vine was the subject of a bid from Cardiff City, believed to be in the region of £650,000, this week, which Mike Newell turned down. It is thought the former Portsmouth man is valued by the Luton boss at around £1million.
The Hatters are not in a position of having to sell, however, as they have already seen Carlos Edwards leave Kenilworth Road for Sunderland's Stadium of Light in a £1.4million deal.
Grant publicly admitted his interest in Vine, having known of him since the pair were in close proximity on the south coast in the Norwich boss' Bournemouth days.