Zwanziger, who was formerly head of the German Football Association, said the plan, still to be ratified by the UEFA executive committee, envisaged 12 countries hosting the expanded tournament which will feature 24 teams, up from the current 16.
"Michel Platini has told me his plan in a personal discussion," Zwanziger told Bild newspaper. "But it still needs to be ratified by the executive committee."
According to Zwanziger the plan foresees 12 countries hosting matches, with the countries chosen as seeded nations based on their UEFA ranking at the time and whether they have qualified for the competition.
If Germany qualified for the tournament they would play their group games at Berlin's Olympic stadium, according to the newspaper.
Platini had said in October he would press ahead with the idea of staging Euro 2020 across the continent only if member associations were in favour of the plan.
The Frenchman said UEFA was waiting for feedback on its proposal, first announced at the end of Euro 2012 in Poland and Ukraine, and it would be a one-off.
"If the idea is appealing to them we will enter into details to see how we can organise a Euro in several European cities but only in 2020 mind you," Platini said in St Petersburg after a meeting of UEFA's executive committee last month.
Platini has said the bigger competition in 2020 will be too costly for single or joint bids.
His idea of having matches in a dozen countries with the final stages in one venue will be discussed by the executive board in December, with a decision made early next year.
Turkey have expressed a wish to stage the tournament on a conventional, single-country basis but their bid will be dropped if Istanbul is awarded the Olympics in the same year.
Scotland, Wales and Ireland have also declared their interest in staging the finals.