The Scottish Championship club have made contact with the Saddlers over the 23-year-old, who is out of contract in the summer, but it is understood the club have rejected their approach and no agreement has been made for the player to leave the Banks's Stadium.
Walsall, who are still targeting promotion from League Two this season, will not be forced into selling players this month and will only let them leave if it is in their best interests, it is believed.
Adebayo, who has netted eight times so far this season, has emerged as the first choice striker this campaign with some dominant displays up front.
Hearts currently sit top of the Championship as they try to navigate a return to the Scottish Premiership.
They are in need of added firepower to propel them to promotion but their approach for Adebayo has been knocked back.