The 19-year-old academy graduate is out of contract with the East London side next summer as a number of clubs - including Southampton and Manchester City - look to capitalise on the uncertainty surrounding his Hammers' future.
But The Mail report that it's Newcastle who are desperate for the England Under-20 international.
The report goes on to state that West Ham rate the youngster very highly, and that the club would be "loathe" to lose him this early in his development.
Of course, West Ham, under Slaven Bilic, look very much like a team on the up, and young players are often the casualty of such lofty ambitions as clubs look for established talent to help them reach a new footing.
Consequently, Burke may feel that his first-team prospects would be better at Newcastle, who were relegated from the Premier League this past season.
Working alongside a manager of Rafael Benitez's vintage is also certainly a draw for young players, given his status within football.
He managed 36 appearances during an impressive loan spell at Bradford City this past season, winning the club's Player of the Year award in an achievement which elevated his status further.