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Francis says Liverpool skipper Jordan Henderson pitched the idea to his fellow Premier League captains.
Donations to the initiative will be anonymous and depend on what players are “comfortable with donating”.
“It’s the power of everyone coming together,” Francis, 35, told BBC Sport.
Francis said top-flight club captains had discussed the idea in group discussions on Whatsapp and Zoom.
“Jordan took the initiative to pitch the idea to the rest of the lads and it was a no-brainer for us,” the defender told BBC sports news correspondent Natalie Pirks.
“It shows so much support within the football community to show we are one.
“We don’t know when that will get up and running, so in the meantime it’s a great chance for players to show how much the NHS means to us – a cause that is close to a lot of players’ hearts.”
Each club will organise their own donations, which will be distributed across the country, though the level of contributions has not been announced.
“It’ll come down to talks with everybody within their own clubs – what they’re comfortable with donating,” Francis said.
“There’s a lot of players that play overseas and have family overseas and they may choose to donate back to where their families are from, which is understandable and absolutely fine.
“The talks are due to take place with each club, Jordan has the fund set up, and then hopefully it’ll be distributed across the country.”
Francis said it was “important” that donations are anonymous, adding: “It’s not about who’s put in more. We are not competing with the top teams, it’s about helping first and foremost. We don’t need it coming out in the press who has done what.”
Meanwhile, Hancock praised the “big-hearted” gesture when it was announced on social media on Wednesday evening.
He had previously said footballers should “do their part and take a pay cut”, which Francis described as “disappointing.”
“I don’t want it to seem like it was a knee-jerk reaction to what has been said in the media – it wasn’t that at all,” Francis said.
“I spoke to Jordan in the morning of the day Matt Hancock came out and made his comments, so it was bad timing.
“With the problems we have across the country, for him to pick out footballers was disappointing.”
More than 150 top-flight players posted a joint statement on social media on Wednesday confirming they were collaborating on a voluntary initiative, “separate to any other league and club conversation”.
Premier League clubs previously said they would ask players to take a 30% pay cut in order to protect jobs,
However, the Professional Footballers’ Association said that would hit tax contributions to the NHS.
On Thursday, Southampton became the first Premier League club to announce their players will defer part of their salaries.
Francis said the Premier League’s request was “hasty” and said discussions at Bournemouth over wage deferrals were “positive.”
Last week, Francis’ manager Eddie Howe was the first Premier League boss to take a voluntary pay cut, with Brighton’s Graham Potter following suit.
What has been the reaction?
West Ham midfielder Robert Snodgrass, who is part of the initiative, tweeted that Hancock should “do your homework on what we do and who we are as people”.
He wrote: “It’s not about us, it’s about the real heroes… the NHS.”
Former Stoke and Burnley striker Jonathan Walters praised the players for establishing their own initiative.
“They do get paid a large wage – I’m not here defending that but as soon as you cut that wage, it’s going to affect the money that’s going into the tax,” Walters told BBC Radio 5 Live Breakfast.
“This initiative is donating directly to the charities that they want the money to go to.
“It’s not as if they’re taking a cut and it’s going back to the owners.”
Former England goalkeeper Rob Green said it was a “brilliant” idea.
“Shock, horror – footballers are nice people and do have the best interests of the community at heart,” Green joked on BBC Radio 5 Live.
“I think it’s brilliant that they’ve done something that works on an individual basis and they’re giving what they can give straight to the NHS.”
Manchester United captain Harry Maguire said it had been “heartwarming” to see the nation come together “to show their appreciation to the NHS and other critical workers”.
Meanwhile, Wolves skipper Conor Coady says footballers “want to make as much difference as possible” during this “horrible time”.
“Footballers are good people,” defender Coady, 27, said.
“We want to try and help the right people at the right times, to make sure that people are getting the best possible care. It’s a horrible, horrible time we’re in.
“It’s about everybody coming together, not trying to point fingers at certain people within the world and trying to say they’re not doing enough.”
The former Liverpool player also praised the “absolutely brilliant” Anfield captain Henderson for driving the captains’ initiative.