Lingard, who left United when his contract expired in 2022 after 20 years at the club, joined Corinthians in January following stints at Nottingham Forest and FC Seoul.
He has been following his former club's remarkable turnaround since Carrick's appointment in January with close interest.
"I think they're really on the right track with Michael. I know him from my United days. I know how he operates. The lads are doing really well under him. He's got that Manchester United DNA inside of him. He knows the ins and outs of the club."
Lingard also weighed in on the Bruno Fernandes Ballon d'Or debate, backing the Portugal captain emphatically after Paul Pogba suggested this week that Fernandes would be considered for the award if he played for a stronger team.
"100%. His performances at Manchester United this season have been extraordinary. He has to be up there."
Lingard settled quickly into life at Corinthians. Former teammate Memphis Depay, already at the club, helped ease the transition. He scored his first goal in the Copa do Brasil last week and is learning Portuguese without a translator.
"I'd never experienced it before — fans going into the training ground, speaking to us. You see how passionate they are for us to do well and win. It only gives us more reason to win on a matchday."
On why he chose Brazil over other offers, he told BBC Sport: "It's still high-level football. I think I can play at a high level. For me, it was just about the competition — how big the club is here, how big the league is. I had offers on the table, but I still test myself out. I've come here to lift a trophy."
On the World Cup, Lingard backed England.
"We've always had a good chance. I think in big tournaments, we always do well. I believe in the lads. I know how good they are, so I don't see why not."