The Kilmarnock full-backs have both been linked with moves to Celtic Park this summer.
Any hopes Celtic had of signing a pair of Kilmarnock full-backs appear to have been dashed.
The Bhoys have lost five senior full-backs - Kieran Tierney, Emilio Izaguirre, Mikael Lustig, Cristian Gamboa and Jeremy Toljan - since the end of last season, and replaced them with just Boli Bolingola-Mbombo and Hatem Abd Elhamed.
But neither Taylor nor O'Donnell - who play on the left and right side respectively - will be allowed to leave for Celtic Park or anywhere before the deadline, according to their manager, Angelo Alessio.
"I want to keep the best players in our team: Taylor, O'Donnell and other players," Alessio said, as quoted by the Scotsman.
"In the next two weeks I don't think this player can go to other teams. I've read about Greg but I think they are just rumours.
"I don't think (the club would sell him) because the club and I want to keep the best player in our team."
Kilmarnock have already said to have rejected a £500,000 offer from Nurnberg for Taylor, who has two years left on his Rugby Park contract.
O'Donnell's deal expires a year earlier, however, which means he will be free to discuss a pre-contract move with other clubs from the New Year unless a new one is agreed by then.
Should Celtic move for Taylor and/or O'Donnell before September?
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