The Portuguese, appointed at the end of December with the Swans in real trouble, was unable to achieve his task of keeping them in the top flight as relegation was confirmed on Sunday.
Widespread reports suggest Carvalhal will be on his way out of the Liberty Stadium in the coming days, but the former Sheffield Wednesday boss has instead claimed that Swansea's American owners want him to stay on.
"The news came out this week, it was not real what people said," he told reporters. "I speak to the owners and they ask me to work next season even if we do not know what will happen. I need time to think, talk with my family and friends, we will talk again and then take a decision.
"This is a fantastic club, the organisation and fans and experience in Swansea means they must be back [in the Premier League] as soon as possible, whether it is with me or not we will see.
"[The owners] said they want to talk to me and that makes me proud. They ask me about the possibility and we will talk. I said we needed two or three days. I will make a decision and so will they."
Swansea's majority shareholders Jason Levien and Steve Kaplan issued a joint statement following the loss to Stoke, admitting that they must take accept "much of the responsibility" for City's relegation.