Markovic has dropped way down the pecking order since his £20 million move to Merseyside from Benfica, and is yet to feature in a competitive game under Jurgen Klopp.
The 23-year-old has already attracted interest from Russian Premier League outfits Spartak Moscow and Lokomotiv Moscow, and now two other sides have entered the race.
This could be either a permanent or temporary switch for the winger, though it is likely the Reds would prefer the former given Markovic’s lack of prospects at Anfield.
Markovic has 17 months left on his current contract, and capitalising on any residual value would be a wise decision this month.
Markovic’s success in the second half of the season with Hull last time out, despite their failure to avoid relegation, may be fuelling Swansea’s interest.
Addressing his desire to add more bodies to his squad in a bid to avoid relegation, the Portuguese acknowledged the need for committed players.
“The reality is the market is not easy, the players are not easy to achieve,” he said.
“I am the leader, I must understand the situation and fight with the players I have.
“We don’t wish to go to the last minute, but our position means we have to stay more to the last minute.
“I explained before, when you are in a position of trying to fight to be in the Premier League, not all the players want to come to the club.
“Because they don’t want to fight in this situation, especially the good players. We must understand that.”
While it may be a back-handed compliment given Carvalhal’s reference to “good players,” Markovic’s 14-game spell with Hull could suggest a move to south Wales would be ideal for all parties.