The 24-year-old Iceland international joined Heerenveen from Belgian club Lokeren last summer, and has hit the ground running in Holland - bagging 16 league goals in 19 games so far this term.
Aston Villa were reportedly monitoring the Icelandic star's progress, but Finnbogason has admitted a move to Manchester United is his ultimate goal.
Speaking to Italian website TuttoMercatoWeb, Finnbogason said: "My biggest dream is to one day become a Manchester United player. This is the club I have been a fan of since I was five years old. I'm still a big fan of United."
The 6ft forward has 13 caps for his country, scoring three goals. He's currently the second top scorer in the Eredivisie, behind Vitesse striker Wilfried Bony.
Bony has 18 goals so far this campaign, compared to Finnbogason's tally of 16, but the Icelandic star is more important to his side, according to statistics.
The Heerenveen star has contributed 48 percent of his club's goals, the highest percentage of any player in the Dutch top flight.
Manchester United face Real Madrid in the Champions League last-16 on Wednesday evening, as Sir Alex Ferguson's side travel to the Bernabeu for the first-leg.