Boro had made their move in the last 24 hours for the highly-rated striker, but their offer of around £4million has fallen short of Genk's valuation.
The 24-year-old, who had also been linked to the likes of Celtic and Newcastle this summer, is now set to stay put.
Genk have now confirmed that there was an offer for Vossen, which sporting director Dirk Degraen described as 'timid'.
"There was indeed today [Monday] a timid attempt," Degraen told Het Nieuwsblad.
"But there was nothing on the table even to be able to negotiate."
Despite their failure to land Vossen, Boro are still hoping for bringing in two strikers on Monday.
One of which is set to be Kei Kamara - who is set to finalise his move from MLS side Sporting Kansas.