Former midfielder Carrick, who made 464 appearances for United, was named interim head coach following the sacking of Ruben Amorim in January.
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The 44-year-old won six of his eight games in charge as the club moved from sixth to third place in the Premier League.
Former Manchester United and England striker Owen, who played with Carrick at Old Trafford between 2009 and 2012, believes he has “returned calm to the club”.
“I can’t believe people are wondering if I should get the job,” Owen said on the latest edition of the BBC podcast The Wayne Rooney Show.
“Manchester United have waited about 12 years; they have looked for proven legends, all the way back to Sir Alex (Ferguson).
“And once they start playing well, getting results, the fans agree. You’re telling me that at the end of the season, if you get third (place), they’ll say ‘thanks, but no thanks’. How is that possible?”
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Rooney, who praised Carrick and previously said the goalkeeper gave United fans a taste of the Fergie era, agrees with Owen.
“Everyone at Manchester United loves Michael Carrick and is simply afraid it won’t turn out well. He’s done an incredible job,” Rooney said.
The club is understood to be looking for other candidates, along with Carrick, to take over as permanent coach in the summer.
The debate between Owen and Rooney
Rooney and Owen were at the center of an intense “who was better?” debate last year.
The conversation began in September when Owen seconded himself in response to a question: “Who was better at 17, himself or Rooney?”
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Owen said injuries had hampered his own career, before adding: “He will become a better player than me. But, at 17, please…”
However, the 2001 Ballon d’Or winner was more diplomatic on Rooney’s podcast when considering who the best footballer was at 18 years old.
Owen recalled the period in which the player who adopted the nickname ‘Wazza’ emerged as a teenage star for England at Euro 2004, six years after Owen caused a similar stir at the 1998 World Cup.
“I felt like it took me a long time to become England’s number one striker and I felt like I was only there two weeks before he came along and I had to become the sideshow,” said Owen, who played 89 times for England compared to Rooney’s 120 appearances.
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Rooney praised his podcast guest.
“I used to play Michael in the 1998 World Cup, and then I was in the team and played up front with him,” Rooney said. “I was in the garden two years before I hit the ball against my grandmother’s wall pretending to be Michael and suddenly I’m playing next to him.”
