Surveys recently went 1.1 million scouts and their families, The New York Times reported.
The Boy Scouts said in January it was considering lifting the ban, and said the issue would be discussed at the national meeting in May.
The survey seeks reactions should the ban be overturned, the Times said.
"If the Boy Scouts of America makes a decision on this policy that disagrees with your own view, will you continue to participate in the Boy Scouts or will you leave the organization?" one of questions asks.
A spokesman for the Scouts said the survey responses would be provided to the national leaders before the May meeting.
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