The BBC understand that Celtic are close to meeting the Rugby Park side's valuation of the player and talks are at an advanced stage.
Neil Lennon admitted earlier in the week that a move for Taylor remained on the cards. "He is an option," he said. "I think it's well documented that we are looking for a left-back. There's been a number of options and Greg would be one of those."
Taylor was left out of Kilmarnock's clash with St Johnstone in Perth with a knock. Should the 21-year-old Scottish internationalist move to the current Premiership champions, he would fight for a starting berth with Boli Bolingoli, after the departure of Kieran Tierney to Arsenal earlier in the summer.