The UK, hit hard by the Covid-19 crisis, this week passed the mark of 25 million jabs delivered as part of a rapid rollout.
Footballers do not fall into the categories vaccinated so far, based on factors such as age and existing medical conditions, but have been travelling for club and international matches since the sport resumed last year.
Southgate feels a strain could be taken off the state-run National Health Service (NHS) if football were able to step in and begin its own vaccination programme.
The England boss, speaking after announcing his squad for World Cup qualifiers, admitted a looming vaccine supply shortfall could prove a stumbling block.
“My view would have been we were close to getting to the point where it would have been acceptable for professional sportsmen to be on that list,” he said. “We are asking them to keep playing.
“They are having to quarantine when they get back from certain situations. They are having to take some risk going back to families and a lot of them have caught the virus because they have been working.
“I was not in any way suggesting they should have been ahead of key workers and teachers… but we are getting close to the points where it could be acceptable and actually, football could afford to save the NHS money by buying the vaccines and administering them.”
Players are set to travel again for international matches this month and some may be forced to quarantine on their return.
England open their 2022 World Cup qualifying campaign against San Marino at home on March 25. They have an away fixture against Albania three days later and host Poland on March 31.