The Premier League side are currently without a boss following last week's sacking of Paolo Di Canio, which came just six months after the Italian was appointed in March.
The 45-year-old, who signed 14 new players in the transfer window, collected just one point from the five top-flight games he took charge of this term.
Kevin Ball was appointed on a temporary basis after Di Canio's departure, and has overseen a League Cup win over Peterborough and another defeat in the Premier League, with Liverpool running out 3-1 victors at the Stadium of Light on Sunday.
However, Byrne has insisted Sunderland officials will not rush into a decision on the next manager.
"Clearly as a matter of priority, our efforts are presently focused on appointing a new head coach," she is quoted as saying in Sunday's matchday programme.
"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."
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.