"It is one of the biggest clubs in the world," Carvalhal tell Sky Sports in this exclusive interview. "Ninety-five per cent of the managers in the world would accept that challenge."
The issue for Carvalhal is this offer has come in the midst of a global pandemic. Moving 5,000 miles from home at a time of quarantine restrictions and fears of another coronavirus spike means uncertainty for a man with elderly parents and two children with diabetes.
"I have a decision to make and I must talk with my family," he reveals. "We must live with the reality and my decision will not just be about football. I will try to anticipate what will happen with the coronavirus in October, November and December. I will try to find the best way to make an important next step in my career but at the same time protect my family."
Carvalhal acknowledges the interest from a club of this stature is not normal. "Flamengo usually contact managers who have won a lot of trophies," he admits. But that is what makes the approach so satisfying - it is an acknowledgement of his impressive season.
Rio Ave, this modest club in northern Portugal, qualified for the Europa League this month, matching the highest league finish in their history and achieving a record points total.
But much more than that, they managed it by playing attractive football too.
"That is why I am really happy," says Carvalhal. "Clubs are coming to me not because of the results but because of the style of play. That is why Flamengo contacted me. I feel that."
From the outset, the agreement with Rio Ave was that Carvalhal would have one good season and then clubs in England would be interested in the former Sheffield Wednesday and Swansea manager. "People said I was either crazy or had a lot of self-confidence."
But what is really fascinating about Carvalhal's approach is that he ripped up the plans that had served him well over a managerial career that dates back more than 20 years.
Coaches at all levels of the game tend to stick to the game plan that has kept them in employment for so long. Those with more modest records than Carvalhal refuse to change.
Yet, this former Premier League manager of the month at Swansea, the coach who guided Sheffield Wednesday to their highest league finish so far this century, recognised the need to update his ideas. At the age of 54, he has decided on a different path.
"I have been doing this since I was 32. Sometimes we need to take some time to reflect. Sometimes we need to do something different. Sometimes we need to do something not because others are doing it so we want to do the same, but because it is different."
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The moment of clarity came during a conversation with former Swansea chairman Huw Jenkins when the team's form dipped as Carvalhal sought to save them from relegation.
"He told me that other teams were blocking our dynamic and he was right. We had started fantastically well and won against Liverpool and Arsenal so they respected us more after that and tried to control what we were doing. We did not surprise our opponents after that.
"There was a similar problem at Sheffield Wednesday. I played 4-4-2 all the time. I remember speaking with my staff and in the third season we felt that it would have been better if we had changed the system or done something different. I had that feeling.
"But we had finished in the play-offs in the previous season, we had the same spine and the players knew the system. I made a mistake. We should have changed something.
"So I had this in my head but I did not change things at Swansea because we did not have too many players. But after this I was thinking to myself that we will prepare a team not based on the system but I will prepare instead a team based around concepts of the game.
"It was Huw Jenkins who made me think a lot about this with some of those questions that he had put to me. It was crucial to changing my mind about the future."
Rio Ave was an ideal place for him to reinvent himself.
"I chose Rio Ave because I had something in my head about wanting to do it differently to how I had done it in my past. Something new. Something that I think will be the future."
"It is a quiet club with good players. I knew Portuguese football and I knew that Rio Ave was a medium-sized club but I also knew they are organised off the pitch. We felt that we could achieve something special playing in a different way.
Carvalhal has used a variety of formations this season, retaining some key principles of play but changing the focus of his training sessions and encouraging his players to think more about the game. The result is a team that has been more adaptable, found more solutions.
"They are thinking more about the game individually and tactically," he adds. "My idea was to prepare a team not about the system but with concepts of the game to get the players thinking about how to find the right way. My staff and I believe we have found that way.
"This season was more of a challenge for ourselves than anything. We put into practice something different. It was a challenge and we won that challenge in terms of results and the very good football that we played. So we are more confident to develop our ideas now.
"We are thinking all the time about how we can upgrade what we are doing. We must think all the time about what we can do better. We cannot just stay the same."
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This willingness to change has clearly paid off. "I have different proposals from different parts of the world," he says. It is just unfortunate the coronavirus complicates matters.
Another reminder that there are more important matters than football came earlier this month when Carvalhal returned from an away game in Madeira in the early hours of the morning and was attacked by three muggers just outside his home. Inside the house, his 23-year-old son was alerted to the danger and they managed to chase off the assailants.
"It was dangerous," Carvalhal admits. "My left and hand and arm were hurt. They had to close the cut with stitches. Physically, I am better now. But the mental side was traumatic and will take longer. It was really bad, especially for my son who went through it with me."
Now his family must sit down and work out what is next.
An adventure in Brazil, perhaps?
"There are a lot of questions that I must put to my family and afterwards we will decide."
Whatever the plan, Carvalhal is confident that his change of approach ensures the future is bright. "Things were really good before," he says.
"But I believe now they will be better and better in my career."
Since speaking to Sky Sports, and for the reasons discussed above, Carlos Carvalhal has taken the decision to make a move much closer to home and has been confirmed as the new manager of Portuguese side Braga.