Jane Land, 37, and Matthew Glover, 46, who are the founders of the movement Veganuary, say the Pope has been selected for his environmental leadership and influence.
They launched campaign Million Dollar Vegan this week to encourage people to eat less meat and dairy in order to fight "climate change with diet change".
The project has been backed by numerous celebrity figures including Sir Paul McCartney, Moby, Woody Harrelson and Mena Suvari.
It is being fronted by Genesis Butler, from California in the United States, who has been vegan since the age of six and staged many protests against animal cruelty.
She kicked off the appeal by sending Pope Francis an open letter inviting him to give up meat and dairy for the Catholic period of Lent - which lasts six weeks up until Easter.
Pope Francis, 82, leader of the world's estimated 1.2 billion Roman Catholics, has previously been outspoken on environmental issues.
"Climate change is a global problem with grave implications: environmental, social, economic [and] political," he said in his 2015 encyclical.
The group hopes the Pope's followers will also change their diet for Lent, saying that if all Catholics took part, it would be equivalent to the Philippines not emitting carbon dioxide for a year.
The Vatican has yet to respond to the request, said Jane who insisted they would pay out the £771,405 if the Pope took up the challenge.
She said: "Our aim to get animals out of the global food system.
"We are fully prepared to pay this money. It would be the best use of this money because of the influence the Pope has in telling people about a plant-based diet.
"The money is in the bank account and ready to go to his chosen charity."
Million Dollar Vegan CEO Matthew Glover, who lives in Copmanthorpe near York, added: "We are launching this deliberately bold, audacious campaign to jolt our world leaders from their complacency.
"For too long they have failed to act on evidence of the damage caused to people and the planet by animal agriculture.
"Worse, many have defended and subsidised that very industry.
"We are thankful that Pope Francis has spoken out on these issues.
"The $1m has been donated by the Blue Horizon International Foundation which invests in companies producing plant-based foods and aims to "accelerate the removal of animals from the global food chain".
The cash offer is part of an international campaign highlighting how animal agriculture "has a greater impact on global warming than the fuel emissions from the entire global transport sector combined".
It also states it is a "leading driver of deforestation and biodiversity loss and causes suffering to billions of farmed animals".
Genesis became the youngest person ever at the age of 10 to give a TED talk, which focused on the environmental benefits of maintaining a plant-based diet.
She said: "Farming and slaughtering animals causes a lot of suffering and is also a leading cause of climate change, deforestation, and species loss.
"When we feed animals crops that humans can eat, it is wasteful. And with a growing world population, we cannot afford to be wasteful."
Jane and Matthew founded Veganuary and dreamt up the idea over a bottle of wine at their Fishergate home back in 2014.
The idea was to launch the Veganuary website to encourage people to give up all animal products every January to highlight animal welfare and environmental issues.
The couple say they used all their life savings in the venture which has been a huge success.
Lent is a period of about 40 days that ends just before Easter and during which many Christians practise abstinence of some kind.
It reflects the Bible story of the 40 days Jesus spent fasting in the desert and in 2019 runs from 6 March to 18 April.
