Express Sport revealed earlier this month that Manchester United and Liverpool are eyeing the Germany international.
Leipzig are keen to keep hold of Werner, 23, until next summer.
However, McGarry claims Liverpool have opened talks over a deal for the striker.
The transfer expert went on to say Werner is keen to head to Anfield to link up with fellow German Jurgen Klopp.
"Timo Werner is a target for Manchester United and Liverpool in the January window but it may be the summer before he actually makes the move,” McGarry said on Reach Plc’s Transfer Window podcast.