Mario Gaspar's scissors kick in last year's Spain against England friendly was also among the 10 candidates along with a winding solo effort from the women's Under-17 South American championship, scored by Venezuela's Daniuska Rodriguez.
Real Madrid and Portugal forward Cristiano Ronaldo was missing, however, as none of the 49 goals he has scored in 2016 were considered worthy of inclusion.
The winner of the award, named after the late Hungary captain Ferenc Puskas, will be decided by the public and announced at FIFA's annual player of the year ceremony on Jan. 9.
Simon Skrabb of Swedish side Atvidaberg scored one of the most unorthodox goals of the goals.
Kristian Bergstrom hit a hard, low cross into the area and, although the ball went behind Skrabb, the Finnish player threw himself forward and manage to turn it into the net with his raised heel.
It was reminiscent of the so-called scorpion save performed by Colombia goalkeeper Rene Higuita in a friendly against England in 1992.
Messi's entry was a sublime free kick he scored in Argentina's Copa Centenario match against the United States while Neymar was included for a Barcelona goal in which the Brazilian flicked the ball over an opponent's head and volleyed home.
FIFA has always preferred acrobatic volleys when choosing candidates but did include Hal Robson-Kanu's goal for Wales against Belgium at Euro 2016 which was as much about the flowing build up from the back as the forward's superb turn and finish.
Mohd Faiz Subri, of Malaysian Super League side Penang, was nominated for a 35-metre free kick in which the ball dipped and swung in a bewildering fashion to leave the goalkeeper stranded.
South Africa's Hlompho Kekana, another unlikely candidate, earned his place after scoring from inside his own half during the Nations Cup qualifier in Cameroon.
Having won possession near the touchline, Kekana spotted the goalkeeper off his line and unleashed his effort from around 60 metres - the 30-year-old's first goal for his country.
(Writing by Brian Homewood, editing by Ed Osmond)
