Sources say the outgoing first lady had planned to light up the executive mansion in the iconic colour scheme in June to mark Pride 2020.
Donald Trump previously voiced his support for Pride and is on the record as having said he is "fine" with the Supreme Court's ruling in favour of same-sex nuptials.
The president subsequently came under fire from the gay community over his decision to remove housing discrimination protections for transgender people.
He also banned trans people from serving in the US Army, a move which was condemned by campaigners.
Two Republicans insiders told the Washington Blade that Mr Meadows had a significant role in ensuring the White House was not seen to be celebrating Pride.
It wasn't immediately clear if Mr Meadows directly intervened in the rainbow projection plans, the outlet reported.
The Independent has contacted the White House for comment.
The last time America's most famous address was lit up with a rainbow was on 26 June, 2015, during the Obama administration.
That came in the wake of the Supreme Court's landmark ruling that allowed same-sex couples to get married nationwide.
At the time, president Obama said: "To see people gathered in the evening on a beautiful summer night and to feel whole and to feel accepted and to feel they had a right to love, that was pretty cool.
"That was a good thing."