Kevin Ball has been handed the reins on a temporary basis following the dismissal of Di Canio in the wake of last weekend's defeat by West Bromwich Albion.
A number of high-profile names have been linked with the role but Byrne insists the right person has yet to be identified and Sunderland are still assessing their options.
Writing in her programme notes ahead of Sunday's Premier League clash with Liverpool, Byrne said: "Clearly as a matter of priority, our efforts are presently focused on appointing a new head coach.
"As you would expect with a position of this magnitude, there has been a lot of interest. However, there is no definitive timescale on an appointment as we are carefully examining all of the options available to us.
"I am sure that you have read and heard all kinds of rumours and speculation about potential targets, but much of this is guesswork.
"We have had contact with some interested parties already and have others to speak to in the coming days.
"As soon as we have identified the right person to take the club forward, we will make an announcement."
Former Brighton boss Gus Poyet remains the bookmakers' favourite after confirming his interest in the vacant post, while Stuart Pearce's odds were slashed on Sunday morning.
Ball has indicated he would like to be considered for the permanent role, and former Manchester United coach Rene Meulensteen also figures prominently in the betting.