Carrick is under increasing pressure after United's 2-0 defeat at Chelsea on Saturday extended their winless run in all competitions to nine matches, their worst sequence in 42 years.
United are just one place above the Premier League relegation zone after collecting seven points from their opening nine matches. They are also 20th in the Champions League standings following a disappointing 4-1 defeat at Como last month.
With Sunday's Premier League match against Aston Villa their final fixture before the international break, Carrick was asked whether he believed he had just two opportunities to save his job.
"No, not at all," the 45-year-old replied. "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 or how we're preparing for the game tomorrow. I'm actually really looking forward to it. I can't wait."
Carrick also insisted that the atmosphere inside the dressing room remained positive despite United's disappointing results.
"The commitment of the players to the club, to me and to each other, I don't question that at all," he said.
"We need to play better and make better decisions, but commitment is not an issue."
The United manager also defended Lisandro Martinez following the Argentine defender's red card in the defeat at Stamford Bridge, saying mistakes were part of football and should not be confused with a lack of commitment.
Carrick will hope Tuesday's match against Roma can provide a turning point for his struggling team, while the fixture also brings back special memories from his playing days under Sir Alex Ferguson.
He scored twice in United's famous 7-1 Champions League quarter-final second-leg victory over the Italian side in 2007.
"It was a magical night," Carrick recalled. "It was one of the best nights, one of the best games and one of the best atmospheres I've been fortunate enough to witness.
"The fact that I scored a couple, which was very unusual, gives me great memories. But it was more the emotion of the night. It was a special one."