The absence of Welbeck significantly increases the chances of Southampton forward Rodriguez being given his debut.
Welbeck and Carrick featured in last month's 2-0 World Cup qualifying win over Poland, but while the former missed United's victory over Arsenal on Sunday with an ongoing knee problem that has kept him out of action since October 19, Carrick played the full 90 minutes at Old Trafford despite a lingering Achilles injury.
No replacements have been called up, leaving manager Roy Hodgson with 26 men to choose from for the Wembley friendly against Chile on Friday, and the visit of Germany four days later.
Hodgson has already described the uncapped Southampton man as an ideal replacement for Welbeck.
Hodgson said when he named his squad: "I see in Rodriguez someone who can fill that Welbeck role.
"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 have been forced prepare for the games away from St George's Park due to a suspected stomach virus.
Hodgson's senior squad were due to assemble at the National Football Centre in Burton-upon-Trent on Tuesday, but instead reported to the Grove in Watford before training at Colney, while the U21s were relocated to Milton Keynes ahead of their Euro 2015 qualifiers against Finland and San Marino.
Captain Steven Gerrard has been troubled by a hip problem for the last couple of weeks and is expected to play just one of the two matches, while Frank Lampard, who will be awarded a commemorative 100th cap before the Chile game, could skipper the side on Friday night.