The 23 year-old currently plays for Schalke 04 in the Bundesliga, and is under contract for another year.
With Klaas-Jan Huntelaar as the No.1 striker, Pukki has struggled to hold down a regular place at Schalke since moving there for £1.2m from HJK Helsinki two years ago.
The tall Pukki caught the attention of the German side when he scored a hat-trick against them in a Europa League tie.
He can operate through the centre or coming in from the left, and is expected to cost around £2m.
Neil Lennon, the Celtic manager, has already lost out in his pursuit of the Latvian, Artjoms Rudnevs, who did not want to move from Hamburg to play in the SPFL Premiership.
Finnbogason - the Icelander played for his country against the Faroe Islands last night but did not score - became the next target on the list of those identified to add firepower to the side which has lost its top scorer for the last three seasons, Gary Hooper.
However, his club side, Heerenveen, are holding out for significantly more than Celtic value the 24 year-old at.
This could open the way for Celtic to make an offer for Pukki, although there are other names on the list which Lennon has collated along with the chief scout, John Park.
The Parkhead club can still add one more signing to their squad for the Champions League play off games against Shakhter Karagandy.
Lennon and his players are scheduled to leave for Kazakhstan straight after their Premiership game against Aberdeen at Pittodrie on Saturday and, with time fast running out to obtain a visa, it's now unlikely that any new signing would be brought on board in time for Tuesday's tie.
To be eligible for the second leg at Parkhead on August 28, any new arrival would have to be registered by Monday next week.