Clark confirmed at a press conference at the team's headquarters yesterday that Joseph would leave the franchise at the end of the Super rugby season to coach Japan.
"At this stage, we have got plenty of time,'' Clark said.
"We don't have to replace the coach until the 2017 season.
"We have got plenty of time and we will be taking that time and we will put timelines and processes in place over the next four to six weeks.'
Assistant coach Tony Brown, who coached Otago from 2012 to 2014, is a potential replacement from within the franchise.
But head coaching positions at Super rugby franchises do not often pop up and there is expected to be plenty of interest in the role.
"Tony will be part of the coaching team. What role he plays in that will be determined over time,'' Clark said.
The Crusaders are also searching for a replacement for departing coach Todd Blackadder next year.
New Zealand Rugby employs the five franchise coaches in the country, but Clark said the franchise would decide who its new coach is.
"There has never been a situation where a Super rugby club has wanted a coach but hasn't been able to get one. So we will make the decision, yes.''
Clark said the search for a new coach would be different from 2010, when he and Joseph met in a coffee shop to start from scratch after a disastrous season.
"In that six years we have built a pretty strong and robust organisation with some really solid people. So we have got a pretty high performing coaching management team now. We will take our time and assess all our options in replacing Jamie,'' Clark said.
Joseph's replacement will become the ninth coach to guide the franchise since Super rugby kicked off in 1996.