Sir Alex Ferguson is believed to be keen to bring the 26-year-old to Old Trafford to further strengthen his goalkeeping department.
Should Muslera complete a move, he would face competition from David De Gea and Anders Lindegaard for a first-team place.
The Uruguay international played twice against United in the Champions League group stage, conceding just one goal and keeping a clean sheet at the Türk Telekom Arena.
“The Premier League is my childhood dream,” he told Haberturk.
“I always want to move forward in football. I started off with Wanderers, then Nacional, then Lazio and then, a step forward, Galatasaray.”
“Whatever game you’re playing in, it gains importance when you play for this club,” said De Gea. “This is a trophy we enter into seriously and want to win.”