The Daily Express reports that United and Liverpool have both sent their scouts to watch the midfielder in action.
Turan recently signed a new contract with Atletico Madrid and has a £33million release clause.
United boss David Moyes is desperate to strengthen his midfield after a sub-standard opening to the season.
Turan’s agent Ahmet Bulut revealed last season that there was interest from United and Liverpool.
“A destination in the Premier League would be to the liking of the player and I can confirm there are clubs interested, like the two mentioned,” said Bulut.
“He wants to work in England.”
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Liverpool have unveiled their new home kit for next season, sticking with the hugely popular retro styling of the current shirt.
The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.