Who would the FA appoint, if they did sack him?
The bookies have former Newcastle boss Alan Pardew in the running.
The possibility of an extension to the 68-year-old's four-year reign has been a simmering undercurrent of the team's time in France, threatening to come to the boil when outgoing Football Association chairman Greg Dyke discussed the minimum expectations for such a deal in a BBC radio interview.
FA chief executive Martin Glenn weighed in on Hodgson's side, offering "total support" and describing him as a "great manager", but the man himself demurred on the eve of the must-win last 16 clash in Nice.
"Every result is significant when you're a football coach, so of course this result is very significant, but more so for the football team and the country," he said.
"We desperately want to stay at the tournament, we think we're good enough to stay, but to do that we've got to get results.
"That's got to start tomorrow because there are no draws available, it's win or lose and we're aware of that.
"We'll be trying our best to win because we want to stay in the tournament for the team and the country."
Hodgson knows his reputation with England hinges on events in the coming days.
Defeat by the smallest nation to ever appear on this stage would surely bring down the curtain on his reign, while progress to a quarter-final with France keeps alive the dream.
It was, then, interesting to hear him ponder the notion of legacy in respect of his opposite number, Iceland's co-manager Lars Lagerback.
He and Hodgson go back many years to the latter's time coaching in Lagerback's native Sweden in the 1970s and 80s.
Yet when the England boss was invited to pass verdict on his friend's influence, it was hard not to see a reflection on his own as-yet-undefined position at home, compared to the revered status he enjoys in continental Europe.
"You never know the impact you have as a coach, that's decided by other people, usually by the people whose country you've been working in," he said.
"I achieved a lot in Sweden over 12 years. Lars is one of a group of people, slightly younger than myself, who I met at the time and he embraced the type of football (fellow Briton) Bob Houghton and I were playing at the time.
"They liked it and took it on board. You develop a special bond with those who liked it and I have ultimate respect for what he's done. His chances of a statue are a damn sight greater than mine."
The truth of that final comment might be put to the test should he manage to deliver England's first tournament trophy since their sole World Cup triumph 50 years ago.
But that cannot happen unless Iceland, the lowest ranked side remaining at 34th on the FIFA ladder, are beaten first.
Only then will a mouthwatering tie with the hosts come about, though Hodgson and company were denied an early scouting session on France due to travel arrangements.
Asked if he had seen Les Bleus' 2-1 win over the Republic of Ireland, he explained: "I'm sorry to disappoint, but we were in the air at the time.
"We followed the first five minutes in the lounge, and then when we got off the plane the final whistle had just blown. If we are lucky enough to get into the quarter-final, we'll have seen that game and our team will be well prepared to meet France.
"That would be an honour, but Iceland have the same ambition as us."