The 18-year-old United striker has burst onto the scene, scoring twice on his debut in the Europa League win over FC Midtjylland before bagging a double in United's 3-2 win victory against Arsenal on Sunday.
Rashford came into Louis van Gaal's side with both Wayne Rooney and Anthony Martial injured but looks set to remain a part of the first-team picture at Old Trafford for the remainder of the season.
The Manchester-born forward, who could not continue his scoring run in United's 1-0 win over Watford on Wednesday, has previously only been capped at under-18 level for England.
But Hodgson, speaking at a Euro 2016 coaches' workshop in Paris, has refused to rule out the possibility of Rashford earning a place in his squad for this summer's finals.
"I would neither rule him in or rule him out, I just hope he can do well," Hodgson said.
"I have been watching Rashford for two years, so I have known about him for a long time. He is in our system.
"He plays for the (England) under-18 team, so we are delighted that a player playing for our under-18s, players who we believe in and who we think have got a bright future, are getting chances in the first team.
"That is wonderful and long may it continue. I hope he has a good end to the season. Most of all, I hope he is allowed to develop as an 18-year-old should and people don't try to put him under enormous pressure.
"Scoring four goals in three games is a great achievement and if he can keep that up, it is great for Manchester United and England.
"Last time I was with the national coaches we were talking about the players, in particular the under-19 squad which is competing for another major tournament and they think the under-19 squad is a strong one and Rashford's name was mentioned then."
While Hodgson has not dampened speculation that Rashford would at least be considered for the European Championship, he had his say on the teenager's sudden emergence into the spotlight.
He said: "My hope for Rashford, and it is a vain hope, is that he will be allowed to develop in peace and it won't suddenly be a question of 'You have scored two goals for Man United and you're going to be the next Cristiano Ronaldo' or every time he has two or three bad games he is a flash in the pan.
"Let's hope the lad at 18 years of age gets a chance to develop in the same way as Raheem Sterling, Ross Barkley, Wayne Rooney, David Beckham, Paul Scholes - you can name as many as you like.
"Let's hope he gets the same chance to develop at Manchester United and hopefully for England as all those guys in the past and he isn't going to be lifted up so high that the inevitable fall will have to follow."
Any hopes Rashford has of making the cut are boosted by the fact Hodgson prides himself on giving youth a chance and he again stressed a player does not have to be playing club football every week to represent England.
"I think we have taken chances and I think clubs have taken chances (with young players)," Hodgson added.
"I have taken chances by giving young players their opportunities and what they have done is taken it.
"We were talking about Tottenham and the young players, they got their chance and they took it - I have done pretty much the same thing over the years.
"Alex Oxlade-Chamberlain was there in 2012, Sterling was in our squad when he was only 18, 19.
"There are lots of players we can point to, Ross Barkley was another one. There are lots of players who we have given a chance to and not been disappointed.
"A lot of my time with the national team has been spent picking players we think are good enough to play for England but they don't make their club side because their club side has a lot of talented foreign players and they unfortunately can't knock them off their spots."
Source: PA