The People says the Yorkshire club's new owners believe the West Ham boss is the man to lead them back into the Premier League.
Allardyce, 58, is out of contract at Upton Park at the end of the season and talks about a new deal are underway, but his future has not been signed or sealed.
The former Bolton, Blackburn and Newcastle boss has shown in the past that he can guide clubs into the top flight and then, crucially, keep them there.
He won promotion to the Premier League with Bolton and the Hammers.
He has also developed a reputation for making sure his teams punch above their weight when they reach the top flight.