Boro has been credited with an interest in Riis from Preston North End this summer.
Boro are looking to add to their attacking options before the end of the summer transfer window.
Preston is adamant, however, he won’t come on the cheap.
And now a Football Insider report claims the fee could be as high as £8million, with three years left on his contract.
The transfer window closes on September 1 and Boro should be busy before then.
Asked about transfers in general, Boro boss Chris Wilder said: “I’m confident we’ll get all four (players) in by the end of the window. Sometimes it can happen quite quickly and sometimes it can be prolonged.
“There’s a huge amount of work going on every day to make sure things happen. There’s no one who just sits on their hands and doesn’t want to make things happen.
“We all want to make things happen but it’s not, as I keep saying, simple.
“It’s not Championship Manager or Fantasy Football where we just click a button and choose our team.
“We are looking for good players, we face competition for players, clubs sometimes don’t want to let them out for the amount we are willing to offer, and so there is an incredible amount of work and negotiation, persuasion and a bit of luck sometimes I think too.
“But we are confident when this window closes we will have the players to fight for.”