The Russian Football Union (RFU) stressed Fabio Capello remains under contract with the national team.
Shakhtar Donetsk claim there has been no approach from Liverpool for in-demand midfielder Henrikh Mkhitaryan.
The 20-year-old joined the Owls over the summer and has already made nine appearances for Dave Jones' side.
Rangers chief executive Charles Green said: "Rangers and Sheffield Wednesday have agreed terms over Rhys McCabe.
"Sheffield Wednesday have acted in good faith and recognised our rights as a club during the process.
"We wish them every success this season and beyond and while we were disappointed Rhys decided his future lies away from Rangers we wish him well for the future."