Finnbogason only joined Heerenveen from Belgian side Lokeren for a bargain £1million in the summer but has blossomed in Holland under coach and legendary Dutch striker Marco van Basten.
His price-tag has soared to £5million amid interest from Liverpool, but Villa appear to be the most keen at the moment as Paul Lambert looks for another bargain.
Heerenveen have a good record for developing strikers to sell on at a profit and may be willing to listen to offers.
However, graduates from the Frisian club have had varying fortunes after leaving with Afonso Alves and Georgios Samaras flopping at Middlesbrough and Manchester City while Klaas-Jan Huntelaar and Ruud van Nistelrooy went on to greater things.
Villa were also interested in last season's Eredivisie top scorer Bas Dost, who left Heerenveen for Wolfsburg in the summer, and has scored nine goals in 15 starts.