Liverpool managing director Ian Ayre flew out to Ukraine this week to discuss the signing of the 24-year-old, who has made 35 appearances for his country.
However, it has now emerged that the Reds face competition from Tottenham for his signature, although Sky Sports understands the Anfield outfit remain in pole position.
Liverpool have yet to recruit anyone this month and missed out on Egyptian winger Mohamed Salah, who left Basel for Chelsea despite holding 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 was hopeful of making "a major signing".
"I think we will hopefully do something over the next couple of days," he said.
Rodgers' plans could still be scuppered again, though, if Tottenham manage to agree a deal with Konoplyanka before the window closes on Friday evening.
Konoplyanka, 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."