Leverkusen had announced the deal late last month, saying it had paid the Argentine international's release clause to River Plate and the 24-year-old forward was eligible to sign with Leverkusen.
"The Argentine football federation, after discussions with Club Atletico River Plate, have refused the release of player Lucas Alario," the Germans said in a statement.
"Bayer Leverkusen have absolutely no understanding for the federation's and the club's actions. The refusal of the Argentines... means Bayer Leverkusen can now request the registration of Alario as its own player with FIFA."
Financial details were not released but media reports put the deal at 19 million euros (£17.43 million).
Alario, who is in Germany and has been training with Leverkusen for several days, had joined River Plate in 2015, netting 31 times in 71 competitive games.
He also helped them lift the Copa Libertadores in his first season with the club.
(This version of the story was refiled to change to "Bayer" in paragraph 4)
(Reporting by Karolos Grohmann, editing by Pritha Sarkar)
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