Rooney made 559 appearances and scored 253 goals in a trophy-laden 13-year spell as a player at the club. He retired in 2021 and has had management spells at Derby County, DC United, Birmingham City and Plymouth Argyle.
Asked if he would consider joining Carrick's coaching department, the 40-year-old said on the BBC's The Wayne Rooney Show: "Of course I would. It's a no-brainer. I'm not begging a job here, by the way.
"Just so everyone knows, if I was asked to go in of course I would. Appointing the manager is the most important thing."
Following his retirement from football in 2018, Carrick stayed on at United as part of Jose Mourinho's coaching staff and was also part of Ole Gunnar Solskjaer's set-up when the Portuguese was sacked.
Carrick, who had a short stint as caretaker boss following Solskjaer's dismissal in 2021, spent two-and-a-half years as Middlesbrough manager from October 2022 to June 2025.
"I think it would be a really good fit," Rooney said. "Michael loves the football club and would step in to do a job if he can.
"He lives and breathes that club - that's what the club needs."
United, who are seventh in the Premier League, face Manchester City on Saturday.