Liverpool have long been linked with a move for Werner, who is widely expected to leave RB Leipzig at the end of the season.
The Germany international has been in fine form once again this season, scoring 18 goals from 31 games across all competitions.
Real Madrid, Juventus and Barcelona have all been tipped to lure the 23-year-old away from the Bundesliga when the transfer window reopens this summer.
Yet transfer expert McGarry has now sensationally claimed that Liverpool are on the verge of agreeing terms with Werner.
He told The Transfer Window Podcast: “This (Werner) is a guy who since he signed from Stuttgart in 2016 has scored 60 goals in only 108 appearances – so he’s got a very good goals-to-games ratio.
“(He’s a) 23-year-old German international, has been interesting Liverpool for a long time.
“As we all know on the Transfer Window Podcast, and our educated listeners know this as well, when a player refuses to sign a new contract it’s usually because he’s got an agreement in place somewhere else.
“My information is Liverpool are very close to agreeing personal terms with Werner.
“Obviously they have been lacking a natural No 9 out-and-out striker for some time.
“Divock Origi hasn’t made any impression, Daniel Sturridge has obviously had his injury problems and has failed to make an impression when he’s come on this season.
“Adding a proper No.9 to their already very potent attacking force is something Jurgen Klopp has been very, very keen to do.
“Whether they win the title or not I think that’s something they will press ahead and recruit anyway.”
McGarry also claims Liverpool are likely to sign Werner in a £43million deal.
“His contract situation is such with one year left, RB Leipzig would be expecting in the region of €80m (£69m) for a player who had more, may two or three years left on his contract,” he added.
“But I think Liverpool if indeed they have either agreed or are getting close to agreeing personal terms, RB Leipzig will probably be looking at getting less than €50m (£43m) for Werner.
“That would be a bargain, a real value signing.”
Should he join Liverpool, Werner is likely to fight it out with Roberto Firmino for a spot in the Reds’ XI.
The Samba star has scored 15 goals from 40 games across all competitions so far this season.