The Super Eagles have been without a permanent coach since Sunday Oliseh quit the post in February.
The Nigerian Football Federation is hoping to be able to draw up a shortlist of candidates by Friday.
"We encourage applications from all highly-qualified persons, whether they be Nigerians or expatriates," it said.
"Applications are welcome until Wednesday, and the Technical and Development Committee will meet on Friday to look through all applications and release a shortlist."
Caretaker manager Salisu Yusuf guided Nigeria to victories over Mali and Luxembourg in two friendly matches in May, but the NFF president Amaju Pinnick has publicly declared his intention to appoint a foreign manager for the country instead.
Nigeria were rocked by Oliseh's exit, brought about by a dispute over contract violations, and they had to play two crucial Africa Cup of Nations qualifiers against Egypt in March under the temporary charge of Under-23 coach Samson Siasia.
The three-time African champions were eliminated from the competition and will not feature at next year's finals in Gabon.
Opinions are sharply divided about what direction the NFF should follow after the country's failure to qualify for the 2015 and 2017 Nations Cup tournaments under local managers.
Nigeria have been drawn in the same African 2018 World Cup qualifying group B as former champions Cameroon, Algeria and Zambia.
The group includes three of the five African countries who went to the last World Cup in Brazil and they will be battling for a single place at the tournament in Russia.
Algeria and Nigeria both got past the group phase in 2014 and were eliminated in the round of 16 while Cameroon, hampered before arriving in Brazil by a player strike over money, lost all three games and conceded nine goals.