Burger King had - the McWhopper - made using both restaurants' patties last week but McDonalds shunned the PR stunt, aimed at raising awareness of the day.
The burger chain said it had received "unexpected" responses from other brands and had already began developing a pop up shop for sale of the burger, which it proposes should be created using a "signature" ingredient from each restaurant.
"Let's make a burger merger", the brand said in the letter. "All we need from you guys is a simple 'yes."
Addressing McDonald's it added: "Our original proposal still stands. McDonald's, please take your time. We're totally at peace with that."
Last week, McDonald's MD Steve Easterbrook said on the brand's Facebook page:
"We love the intention but think our two brands could do something bigger to make a difference. We commit to raise awareness worldwide, perhaps you’ll join us in a meaningful global effort?
"And every day, let's acknowledge that between us there is simply a friendly business competition and certainly not the unequalled circumstances of the real pain and suffering of war.
"We’ll be in touch. Steve, McDonald’s CEO. P.S. A simple phone call will do next time."
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