Press Association Sport understands the club are in conversation with their Ukrainian counterparts to ascertain whether a deal can be done this month.
Sources have said Liverpool are "testing the water" to see if Dnipro are willing to sell now as the 24-year-old winger was initially identified as target for the summer.
Contrary to reports the Reds head of recruitment Dave Fallows has not travelled out to Dnipropetrovsk and managing director Ian Ayre is believed to be handling negotiations, although not in Ukraine.
However, with less than two days remaining in the January transfer window it may prove a task too far to push through a deal which would likely cost Liverpool £16million.
The Reds have been given some hope it remains a possibility after the Ukraine international's father Oleg said there was a potential deal in the offing.
"I do not want to say specifically but the information about a transfer to the English club is true," he told Ukrainian media outlet Tribuna.
"He can change club later this week. We do not consider offers from Ukraine."
Having lost out on long-term target Mohamed Salah after the Egypt international opted for Chelsea, Liverpool manager Brendan Rodgers said he was still keen to make an addition before Friday's deadline.
There has been growing discontent among fans that no progress was being made to strengthen a squad which is severely depleted by injuries despite being in a battle to secure a top-four finish.
"I think we will hopefully do something over the next couple of days," said Rodgers after beating Everton 4-0 in Tuesday's 222nd Merseyside derby to maintain their grip on fourth place in the Premier League.
"Hopefully we can have a major signing before the window shuts and if we can do that we will be set up for the remaining games."
It was felt Liverpool needed greater reinforcements towards the back end of the team with four defenders - Daniel Agger, Jose Enrique, Mamadou Sakho and Glen Johnson - and defensive midfielder Lucas Leiva all currently out injured.
However, having kept a second successive clean sheet centre-back Martin Skrtel believes they are strong enough defensively to cope with injuries.
"We have been working every single day on the training pitch to try to improve our defence," he said.
"It's not only about the keeper or the back four, it's about the whole team defending better. Last night was a big step forward.
"It's always nice to keep a clean sheet but the most important thing is always to win. I would rather we won 6-4 than drew 0-0.
"With the FA Cup win (against Bournemouth) too that's two clean sheets in a row and hopefully we can make it three at West Brom on Sunday.
"If we keep performing like that then we can finish in the top four.
"There is a four-point difference between us and Everton now, and there aren't too many games left to be played before the end of the season."
With Lucas out for probably another six weeks Steven Gerrard has been asked to play as the deep-lying midfielder and he put in a masterclass against the Toffees, leading from the front to score the opening goal.
"His influence in these games has been wonderful but it was equally as good (on Tuesday) albeit from that deeper position," he said.
"You saw his responsibility in the role, organising the team on the field and scoring a goal."
Everton manager Roberto Martinez knows he has to pick his side up from a demoralising defeat.
"Unfortunately for us it is a performance which came at the wrong time. We need to get everyone back and fully match-fit," said the Spaniard after fielding the likes of Antolin Alcaraz, Steven Pienaar and Ross Barkley who were clearly not ready after several weeks out.
"You shouldn't be too negative about what effect it is going to have in the games ahead of us.
"It is the opposite, it is a real test to regroup together and get ready for Saturday.
"I never thought the result would give a clear indication of what is going to happen in May, the same way we could've won and still not qualify (for the Champions League)."
There were reported incidents of coins being thrown at striker Luis Suarez during the derby at Anfield and Merseyside Police confirmed they were investigating and had requested CCTV footage.
The Football Association is aware and will contact both Liverpool and Everton for their observations but it is a matter the governing body will leave in the hands of the police and both clubs.