By Graeme Bailey Follow on Twitter
The 27-year-old has long been linked with a move to the Premier League, and the Black Cats are hopeful of securing a deal.
Sources close to Dynamo have confirmed to Goal that talks are ongoing between the two clubs.
"The club do not want to lose him, but everyone knows he is going and it is just a matter of price, which is being talked through now," the source said.
Lens joined Kiev in the summer of 2013 from PSV, where he played under current Sunderland boss Dick Advocaat for one season before they both departed Philips Stadion.
Advocaat remains a huge fan of the forward, who has 29 caps for his country, and the club are now hopeful of finalising a deal.
Meanwhile, the north east side believe they are closing in on a number of breakthroughs in the transfer market.
They are also in talks to land former loanees Sebastian Coates from Liverpool and Stewart Downing from West Ham.
It is hoped that recruiting players of that calibre will help persuade another of their top targets – PSV's Georginio Wijnaldum - to move to the Stadium of Light.