Taylor has already rejected a new deal at Boundary Park, stating his wish to play at a higher level than League One.
Lee's performances at full-back this term have earned him Latics' Player of the Season gong and the 23-year-old is currently mulling over an offer to extend his stay beyond the current campaign.
"We have made an offer to Kieran and are waiting to hear from him," Dickov told Oldham's official website. "It is no secret that clubs are looking at him, but Kieran enjoys it here.
"Chris knows that we want him to stay and we have made him an offer, but there are Championship clubs looking at him and talking to his agent.
"Nobody can begrudge him wanting to play at a higher level. If he stays in League One I am confident it will be with us."