The exhibition, concert and lecture hall will be built across the road from the late trumpet ace's modest house in Queens, New York.
With an estimated budget of around $9 million (£4.7 million), the project will become the permanent home of the existing Armstrong archives and exhibits, which have been kept in the basement of the house since 2003.
The trumpeter, nicknamed Satchmo because of his satchel-shaped mouth, lived in the building in 107th Street with his wife Lucille from 1943 until his death in 1971.
However, the popularity of the Armstrong House Museum has meant it has outgrown the current site - the garage has had to be converted into a gift shop - and the organisation's chiefs are hunting for a designer to draw up plans for a new 790m 2purpose-built centre.
New York State has already pledged to fund more than half the construction costs and City University, which owns the proposed site, is expected to come up with the rest.
Michael Cogswell, director of the Louis Armstrong House Museum and Archives said: 'We'll be able to provide more to our visitors and that will bring more of them in.'
Any British architect wishing to enter the competition will have to team up with a registered New York practice.
Once a winning design is chosen, building work is expected to take just one year, and the museum hopes to open its doors to the public in December 2008.
Visit www.nyscr.orgfor details of the competition.