Markovic began pre-season as part of Jurgen Klopp‘s squad, featuring in the 4-0 victory away to Tranmere Rovers at the beginning of July.
The Serbian has since dropped out of contention, however, left out against Wigan Athletic and, subsequently, touring squads for both Hong Kong and Germany.
While he remained on Merseyside to continue training when his team-mates were in the Far East, Markovic has now travelled to Italy, as negotiations continue with Fiorentina.
According to Goal‘s Melissa Reddy, the Florence side are prepared to firm up their long-held interest in the 23-year-old.
Markovic is free to leave Liverpool this summer having fallen out of favour under Klopp, spending the last two seasons away from Anfield on loan.
Stefano Pioli’s Viola have already been busy this summer, signing 10 players so far including Aston Villa duo Carlos Sanchez and Jordan Veretout, but are still lacking wingers.
“With Lazar, it’s no secret that if he finds a club then we will negotiate. It was like this last year too,” he said.
“He’s a really good player. I like him as a person. He’s now fit and another one who is in the best shape since I’ve known him.
“Before he had problems with his groin but now he’s good. There will be a club out there who needs him.”
Having impressed in the second half of last season on loan with Hull City, Markovic has proved he is still capable of shining at a top-flight side.
Liverpool were initially said to be seeking a £20 million fee for the winger, but this was rumoured to have dropped to £12 million, and then most recently to £8 million.
However, Reddy’s suggestion that the Reds are hopeful of reaching an agreement worth £16 million is hugely encouraging.
Markovic arrived from Benfica in 2014 as one of seven signings to break the eight-figure mark at Anfield, but his trajectory has been more Mario Balotelli than Adam Lallana.