Heerenveen face an unexpected exit from the Eredivisie's UEFA Europa League play-offs after they lost the first leg 3-0 on Wednesday, with Finnbogason injuring his knee in Alkmaar.
The Icelandic striker scored 29 goals in the Dutch top flight this season but lasted just one half at AZ in the first leg, as the hosts struck three times before half-time.
"It looks a sprain of the inner knee," Heerenveen physio Erik ten Voorde told the club's website on Thursday.
"(A scan on) Friday should show what is really going on but Alfred can't play Saturday."
Heerenveen entered the Europa League play-offs after finishing fifth in the Eredivisie's regular season - the highest of the four clubs involved - but will have to overturn a three-goal deficit at Abe Lenstra Stadion on Saturday.
AZ's performance in the first leg was a big shock as Dick Advocaat's men had failed to win in eight games in all competitions before playing Heerenveen, while the latter had won four straight matches.
After the first leg, Heerenveen head coach Marco van Basten conceded it would be difficult for his team to recover.
"We are not without a chance," the 49-year-old said.
"On Saturday, we must make it terribly difficult for AZ."
In the other play-off semi-final, Vitesse Arnhem head home after losing the first leg in Groningen 1-0.
Vitesse have not won in eight matches but held on until the 88th minute in Groningen before Lorenzo Burnet struck for the hosts.
Burnet's goal was enough for Groningen to win their seventh consecutive game and they will hope to maintain their momentum at Arnhem's GelreDome, after storming into the Europa League play-offs.
The winners of the two semi-finals will play-off for a spot in the qualifying rounds of next season's Europa League.
