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.

North Wales Police had 473 complaints in 2014/15 - a 43% rise on the previous year - Independent Police Complaints Commission figures (IPCC) show.
Gwent Police had 398 complaints - up 28 - and South Wales Police saw a 20% rise to 864. But Dyfed-Powys had 268 complaints - a decrease of 18%.
There were record number of complaints against police in England and Wales.
A total of 37,105 complaint cases were recorded in 2014-15 - a 6% rise in the amount of complaints made in 2013-14.
The most common complaint made was for "neglect or failure in duty".
In Wales, the number of complaints in the last five years have increased at a slightly higher rate than the increase nationally.
Compared to 2009/10, the number of complaint cases recorded by Welsh forces in 2014/15 increased by 10%. Over England and Wales as a whole the increase was 8%.
The figures found:
The IPCC said the figures also revealed "marked inconsistencies" in whether forces investigated most complaints formally, or used more informal "local resolution" processes.
Dame Anne Owers, chairwoman of the IPCC, said overall the figures showed a complaints system that was "both over-complex and inconsistent, and is clearly failing to satisfy a significant number of complainants".
She said: "Chief officers and police and crime commissioners should look closely at the figures for their own forces to satisfy themselves that complainants are being treated fairly and well.
"However, the underlying problem is the system itself. We welcome the fact that the government proposes to bring in legislation to simplify and streamline a system that at present satisfies neither those who need it nor those who have to operate it."
The number of complaints made to the watchdog marks the most it has received since it started collecting data in 2004-05.
Det Supt Richard Lewis, Dyfed-Powys Police's head of professional standards department, welcomed the reduction in complaints, adding: "It is important to us that people have the confidence to report complaints, and know that we have a thorough procedure for investigating and monitoring complaints."
Nigeria have started the process of finding a new manager of the national team, according to the country's football federation.
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Three Welsh police forces have seen huge rises in the number of complaints they received last year.