The Arsenal winger suffered knee medial ligament damage in Wednesday's 2-2 draw with Ecuador, though the Three Lions are hopeful he will not be ruled out of the whole tournament in Brazil.
|9/1||England are 9/1 with Paddy Power to beat Honduras 3-0 on Saturday|
And boss Hodgson, who previously revealed that the 20-year-old will stay with the squad even if he isn't fit to play during the group stage, has admitted a final decision will not be made on his fitness until June 13 - 24 hours before England kick-off against Italy.
"[There are] no changes," he told reporters. "We will monitor the situation and keep our fingers crossed."
"We will make a decision on Alex at the very last minute. We are working hard on him and he is very positive."
Hodgson's side take on Honduras on Saturday in their final warm-up match before travelling to South America, with a number of players hoping to catch the eye ahead of the competition beginning in earnest.
But the former Liverpool boss revealed he is unlikely to select the eleven that will start against Italy as he does not want to reveal any plans to the opposition ahead of the match in Manaus.
"The players who didn't play on Wednesday have got a very good chance of playing in this game [against Honduras].
"The headache the manager has is that every player is capable of playing in this starting line-up.
"There is no reason for me to give Italy our starting line-up ten days in advance. Against Honduras it will be more than an 11-player game."
One member of the squad who will not take part against the Central American outfit is Raheem Sterling after he was sent-off against Ecuador, but Hodgson has refused to rule him out of contention for a starting berth in Brazil.
The Liverpool teenager is yet to start either of England's warm-up matches against Peru or Ecuador, but his performances in training are suggesting he could still have a major role to play.
"Raheem Sterling's sending off against Ecuador won't affect his chances of starting against Italy.
"He was unstoppable and breathtaking in training today."