"We'd like to get one for the first World Cup qualifier (in Slovakia on September 4), but if we won't we have an interim plan in mind", Glenn said.
All that Hodgson wanted to say about his team's 2-1 loss to Iceland the previous night in the European Championship round-of-16 had been said, he felt, in his resignation statement immediately after the defeat that sent England out of a major tournament early, yet again.
"That's been an issue for many years".
The Northern Irishman was appointed as the new manager of Celtic only at the end of last month, and is exclusively focused on the Scottish Premiership club, who are continuing preparations for their upcoming Champions League qualifier with a friendly against NK Celje in Slovenia.
"We'll see", he said, when asked of the timescale.
"I don't really know what I'm doing here", Hodgson said. Absolutely. So would you rule him out?
The former Liverpool manager still has plenty of admirers within the FA, with the title-winning Leicester City manager, Claudio Ranieri, also likely to be considered.
England Under-21 boss Gareth Southgate could take temporary charge while FA chief executive Martin Glenn, who this week conceded that he was "not a football expert", canvasses opinion within the game.
Gareth Southgate, the England Under-21s head coach, is well positioned to step up to senior duty for the time being with Glenn admitting during a hastily organised press conference on Tuesday that no "perfect" English candidates exist. I don't think we are ruling out anything.
'You need someone who knows about English football. That is what you would look for.
"I'm no longer the England manager, my time has gone, but I was told it was important that I appear here".
The 45-year-old has not been offered the role or spoken to the Football Association since the weekend. If you said: "'This person is an absolute shoo-in, can you wait?' then we are well placed with an interim solution".
Captain Wayne Rooney denied reports that the players lost faith in their boss, and Hodgson himself has insisted that he was unaware of a revolt.
While Iceland will be remembered for its superb and certainly courageous show of force, the fallout for the English National Team has already resulted in the resignation of Hodgson. I think he's got the personality, the strength, he's a good technical manager, he's very experienced and he's someone who perhaps could imbue confidence.