The duo have been linked with moves away from Rugby Park this summer.
Taylor is a reported target of Celtic following the departure of Kieran Tierney to Arsenal, while they rejected a £500,000 bid from German side Nurnberg.
However, Alessio has vowed to keep the player, who won his first Scotland cap earlier this summer, at the club.
Alessio said: "I want to keep the best players in our team: Taylor, O'Donnell and other players.
"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."
While Taylor is contracted to 2021, fellow Scotland international and Killie colleague O'Donnell sees his contract expire at the end of this season.
The Rugby Park side continuing discussions with the player over extending contract.
The Italian manager wants to add to his squad before the transfer window closes at the start of next month.
He said: "We need to add other players up front but this is a very difficult market because other teams want the same players. Maybe next week we'll have some players."