Liverpool have already started negotiations with the Ukrainian club and are confident of completing the signing by Friday.
The club's head of recruitment Dave Fallows is understood to be travelling to Ukraine to hold talks with Konoplyanka.
The Reds have yet to recruit anyone this month and missed out on Egyptian winger Mohamed Salah, who left Basel for Chelsea despite holdings talks about a move to Anfield.
Following his side's 4-0 victory over Everton in the Premier League on Tuesday, manager Brendan Rodgers said that he is hopeful of making "a major signing", and Konoplyanka could be the player who bolsters his squad for the second half of the season.
"I think we will hopefully do something over the next couple of days," he said.
"I saw some headlines after Monday's press conference in terms of the manager being angry with the club - nothing is further from the truth.
"We are not just going to get the players we want to get in for a number of reasons.
"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."
Konoplyanka, a 24-year-old Ukraine international who scored against England in a 2014 World Cup qualifier at Wembley in September 2012, can play on either wing as well as through the middle.
The player's father Oleg has also increased speculation about a move to Anfield by telling Ukrainian media outlet Tribuna: "I do not want to say specifically but the information about a transfer to the English club is true.
"He can change clubs later this week. We do not consider offers from Ukraine."