Club legend Miller said yesterday that the new boss needs to be "at the helm now".
Aberdeen are on a dreadful run of form which has left their top six hopes hanging by a thread in a season which presented an unprecedented opportunity to the club.
But recent addition to the AFC boardroom Yule believes that while time is a factor, the club will not be rushed into replacing retiring gaffer Craig Brown.
Yule said: "Willie's view is a fair and reasonable one. But what we won't do is allow our choice of manager to be determined by haste.
"It is important that we appoint someone as soon as we can. But it is more important we appoint the right management team.
"There has been keen interest in the post. The board is already moving quickly to condense the list of applicants down into what you would best describe as a shortlist.
"Craig's decision to retire meant that we had to start looking for a replacement straight away.
"Now that it is in the public domain it gives us the opportunity to be more open when it comes to making approaches to anyone we might like to bring in."
Yule also explained the thinking behind Brown's new role as a non-executive director at the club.
He said: "We feel Craig can continue to be an asset to the club. He is a real gentleman with a very broad list of contacts throughout football.
"We wanted to create a role for Craig that would allow us to benefit from his strengths.
"His role will include playing a key part in helping us to establish closer links with clubs in England and overseas."
Source: Aberdeen Mad