The report claims Shamrock are expecting interest to step up in the coming weeks and hope that his recent international breakthrough will invite bids of around €250,000 (£219,000).
24-year-old Burke came through the ranks at Aston Villa, and later had a spell at Nottingham Forest's city rivals Notts County, before returning to his native Ireland in February 2017.
However, Burke's 18 goals in 43 games since then appear to have alerted fresh interest from England, as well as the attention of his national team manager, Martin O'Neill, who handed him his debut last week.
Burke followed that up with a goal in his second appearance for the Boys in Green, as they beat the USA 2-1 in Dublin this weekend.
As well as Forest, Blackburn and Brentford, who will all compete at Championship level next season, Preston North End have also been credited with interest in the Dubliner.
And he could find the prospect of a move to Preston appealing, having seen fellow Irishmen Sean Maguire, Daryl Horgan, Andy Boyle and Kevin O'Connor make similar moves in recent seasons.
Would Burke be a good signing for Nottingham Forest, Blackburn Rovers or Brentford?
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