Everton are reported to be closing in on Cenk Tosun after holding talks with Besiktas president Fikret Orman in London.
Allardyce confirmed earlier in the week that they had done all they could to seal a deal, but it looked as though it would not be enough as the Turkish side held out for more.
A deal looked to be dead, but now it is claimed Everton have agreed to go a bit higher to land their man after meeting with Orman in London on Wednesday.
The Toffees have agreed to pay £27million for the 26-year-old and it is now hoped a deal will be pushed through in the next couple of days with Tosun also in the Capital waiting to undergo a medical.
Everton have not replaced Romelu Lukaku, who left for Manchester United in the summer, and new boss Sam Allardyce immediately identified Tosun as the man he wanted to fill the void.
Tosun has helped Besiktas reach the last 16 of the Champions League, where they meet Bayern Munich, and it does come as something of a surprise that the Turkish side are willing to let him go.