hairman Huw Jenkins has been conducting negotiations this week in a bid to find the right man to replace Rodgers, who joined Liverpool earlier this month.
Press Association Sport understands Swansea hope to finalise a deal which will see the successful candidate appointed within the next 24 hours.
If it is to be Laudrup, his compatriot and former Swansea manager Jan Molby believes the former Brondby, Getafe, Spartak Moscow and Real Mallorca boss will be an ideal fit in maintaining the style of football developed by Roberto Martinez and then continued by Rodgers.
"I think he'll do well. I know Michael was very impressed with the way things are done (in the Premier League), the way in which he is in control of footballing matters, selling of players, buying of players," Molby said.
"He was strongly influenced by Johan Cruyff when he was at Barcelona in the late 1980s, early 90s. That's how he wants to play his football, that's how Swansea played last year under Brendan Rodgers, so if it happens I think it's a marriage made in heaven.
"They've put a squad together capable of playing exactly like they want to play. They're brave and it's worked well for them, they've proved they can do it in the Premier League.
"That's very important for Michael, he won't have to start from scratch with a group of players that maybe aren't comfortable having the ball, being in possession. This group of Swansea players are. He will maybe add one or two players he might know from Europe but of course there won't be great changes, because they don't need to.
"I think Michael is a bit like Brendan Rodgers, likes a settled squad, likes to work with the players, feel that he can improve the players. So I don't think he's one of those managers who is going to be knocking on the chairman's door every other day saying they need more players."
Laudrup won five Spanish league titles with Barcelona and Real Madrid in the early 1990s and was one of the players of his generation, but has yet to replicate that success as a manager.