Januzaj, born in Belgium to Kosovar-Albanian parents, has quickly become one of the most sought-after players in world football following a bright start to his senior career at Old Trafford.
The 19-year-old is wanted by England, Serbia, Albania, Turkey, Croatia and Belgium after his seemingly effortless transition to top-flight football.
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Under Fifa rules, a player has to have lived in a country for five years before he can represent them.
But Kosovo are hoping to lure the winger for their first international friendly against Haiti on March 5, even if it just a one-off appearance.
"We are in discussions with Adnan's family and to have a player of his amazing talent play in our very first match would be hugely symbolic," said Eroll Salihu, the secretary general of Kosovo's Football Federation.
"We feel he owes it to his people to take some part in the match, even if it is for just 15 minutes.
"I stress this is a symbolic gesture. By playing in this match we are not thinking he is making any commitment to us.
"It is a friendly and would not have an effect on him being able to play for another country in the future.
"It is not our purpose to do that, but we hope he can be with us on a very special and memorable occasion."
Januzaj is eligible to represent Belgium having been born in the capital, Albania through his parents and Turkey through his grandparents, while Serbia is also an option due to the fact that Kosovo's independence is yet to be recognised by the United Nations.