"It is obviously challenging times but that's not the view of the board [about Pulver]," Clyne said.
"He has certainly not said that [he will step down] and the board has no intention of asking him to go."
The ARU are meeting next week to discuss the process of removing one of their five Super Rugby teams.
Nothing has been decided.
Pulver has faced legal threats from Rugby WA and Melbourne Rebels, with former Wallabies coach Bob Dwyer and player Brendan Cannon calling for his exit.
Castle spent six years in charge of Netball New Zealand before moving across the Tasman to become the first woman to lead an NRL franchise.
Clyne said the board had considered the ARU leadership for next year, but there were more pressing issues to address first.
"With Bill we have a conversation closer to that time [February] but there is no movement at this point," Clyne told the Daily Telegraph.
"Any new CEO you would want to be set up for success and I am not sure the governance structure at the moment is there for that."