Shaw fractured his leg in two places during United's Champions League game against PSV in September and his chances of being picked for the showpiece event in France appeared to have diminished further when he was not called up to an England get-together in February.
The 20-year-old has not played since sustaining the serious injury at the Philips Stadion but, having been included in United's Europa League squad for the latter stages of the tournament, Hodgson insists the former Southampton player is still in his thoughts.
"We are still monitoring Luke Shaw," Hodgson said at a Euro 2016 coaching workshop in Paris.
"He is making good progress and we are also monitoring Jack Wilshere, who is making very good progress. We have to accept that in March we won't have those players [for the friendlies against Germany and Netherlands], but hopefully in May, when I have to make a decision, they will be available for selection.
"The most important thing is the players I select in May are 100 per cent and fully fit to go so I don't have to pick a player who is not fit at that time but may be in two or three weeks' time."
England's expectant supporters will go into the tournament with high hopes despite their team's status as outsiders.
Hodgson added: "We want to do the best we can. Obviously winning also. However, hope and reality are two different things. When you hope, everything is possible. When you dream, everything is possible. Expectations are different things.
"Our expectations are going to be to play well, to play as a team, to perform in an acceptable and correct way. We let others have different expectations and also some hope.
"The greatest thing in such a tournament is that everyone is allowed to dream and hope that their team does something special. Why not England?"