The 20-year-old attacking midfielder has been heavily linked with a move to the Premier League this summer, but could now complete a surprise switch to Ukraine.
Shakhtar manager Mircea Lucescu is on the lookout for replacements for Fernandinho and Willian, who both left the Donbass Arena earlier this year, and has identified Bernard as a man to fill the creative void.
Spurs boss Andre Villas-Boas has also been credited with an interest in bringing the Brazil international to White Hart Lane, as he too looks to strengthen his squad ahead of the 2013/14 campaign.
Atletico Mineiro are reportedly holding out for closer to €25million for Bernard's signature, but are understood to be giving serious consideration to Shakhtar's offer.
With the club also set to lose Armenian midfielder Henrikh Mkhitaryan, Lucescu will be hoping to tie up a deal for Bernard before Tottenham have chance to make a rival bid of their own.
Bernard is one of a trio of young Brazilian stars that Villas-Boas has been casting his eye over, with long-term target Leandro Damiao and Corinthians midfielder Paulinho also on the north London club's radar.