The Southampton forward was called-up for the first time by the Three Lions last week ahead of their double-header at Wembley, having been in good form so far this season that has helped Saints to third in the Premier League.
Manchester United's Welbeck has been ruled out of international duty with a knee injury, and Hodgson has talked up Rodriguez as the perfect replacement.
"I see in Rodriguez someone who can fill that Welbeck role," Hodgson said in a press conference.
"He's a wide forward rather than a midfielder or a winger. He's a wide forward who likes to go forward and attack people, with or without the ball.
"We think he's played well for Southampton and this might be an interesting opportunity for us - 10 days of training and a couple of games - to know more about him than we do now."
England host Chile at Wembley on Friday.