Englishman Redknapp, 65, sacked by Spurs in June, emerged as favourite when Ukrainian Football Federation (FFU) chiefs met on Tuesday, a statement said on the federation's website.
"The praesidium took the decision to begin talks with Redknapp's representatives relating to him working in the post of chief coach of the national Ukraine team," the statement said.
"He has a lot of experience, coaching talent and knows how to motivate footballers," Fomenko said.
"In whatever team he led Redknapp very quickly improved the quality of the play and achieved significant progress. Today he is an expert in the game and one of the best England coaches," Fomenko said.
Ukraine's soccer chiefs first turned to retired striker Shevchenko to fill the post left vacant by Oleg Blokhin who departed to coach top Ukrainian club Dynamo Kiev.
But Shevchenko, 36, who played for AC Milan and Chelsea as well as Dynamo Kiev in a glittering career, turned the job down on Monday saying it was "premature step" for him.
