The Daily Mail says an £11million deal had been agreed in principle between Chelsea and the Ukrainian club for the Brazilian midfielder, but Metalist are understood to have demanded more money after City made their own move for Taison.
Benfica and rival Ukrainian club Shakhtar Donetsk have also made enquiries.
However, Chelsea remain determined to sign Taison and have reopened talks with Metalist chiefs over an improved offer.
Metalist are believed to want an extra £2m from Chelsea and want the new deal agreed by tomorrow.
If Chelsea refuse to stump up the extra cash, Metalist will explore interest from the Premier League champions.