Carrick's position at Old Trafford is increasingly precarious following Saturday's 2-0 defeat at Chelsea, which extended United's winless run in all competitions to nine games - their longest such run in 42 years.
The club are just one place above the Premier League relegation zone after taking a mere seven points from their nine matches so far this season.
And they are 20th in the Champions League table following a dismal 4-1 defeat away at Como last month. Only the top 24 teams will progress beyond the league phase.
Former United and England midfielder Carrick was asked at his pre-match press conference on Monday whether he felt he had just two games left at the helm - the match against Roma and Sunday's clash with Aston Villa.
There is an international break after the Villa game.
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"No, not at all," said the 45-year-old, in his first full season in charge of United. "If you look into the future, you never know what's going to happen.
"I don't know what's going to happen. That doesn't change the way I feel, how we're preparing for the game tomorrow.
"I'm actually really looking forward to the game, can't wait for the game tomorrow. You might think I'm crazy with the results we've had and what's going on, but I genuinely can't wait."
Carrick said the dressing room was in "a good place" despite the terrible start to the season.
"Perception is a funny thing. You can make things look all sorts of different ways and think about all sorts of different things," he said.
"I've said it before, the commitment of the players to the club, to me, to each other, I don't question that at all. We need to play better, we can make better decisions, but commitment is not a thing at all."
He defended United's Argentine centre-back Lisandro Martinez, who was sent off in United's defeat at Stamford Bridge.
"Things happen in football, sometimes you do things, sometimes you make decisions, sometimes you make mistakes, sometimes it goes well for you, but the commitment is not an issue for sure."
Carrick has happy personal memories of facing Roma as a member of Alex Ferguson's team, scoring twice in United's 7-1 quarter-final second-leg triumph in the Champions League in 2007.
"It was a magical night, I have to say," he recalled. "It was one of the best nights, one of the best games, one of the best atmospheres I've been fortunate enough to witness.
"The fact that I scored a couple and that was very unusual gives me great memories, but it was more the emotion of the night, I have to say, it was a special one."
