The veteran Serbia international recently announced that he is set to leave Old Trafford at the end of the season after an eight-year stint.
Vidic has been an influential character at the club for some time, and Moyes is keen to replace his captain with a fresh face over the summer.
Kaya has emerged as a target, with the Gala ace having established himself as one of the stars of Turkish football.
The Istanbul side are believed to be looking to keep the 22-year-old, but it is thought that an offer of around £12m may be enough to lure him away.
Moyes is set for a hectic transfer window in a few months’ time, with the Scot looking to draft in a number of new players.
As well as a centre-back, a top class holding midfielder, a winger and a left-back are also on his radar.
Cheick Tiote of Newcastle has been mooted as a target, but the likes of Ilkay Gundogan, Marco Reus and Toni Kroos are said to be the preferred signings.