The club are looking for a permanent successor to Jose Mourinho who was sacked earlier this month.
Caretaker boss Ole Gunnar Solskjaer will guide the club until the end of the season when United will appoint a permanent successor.
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Tottenham’s Mauricio Pochettino is the current frontrunner, while Juventus chief Massimiliano Allegri and Zinedine Zidane have been linked with the post.
And White says he spoke to Woodward who told him he was not proud of sacking Mourinho but claimed there had been a lot of interest in the job since his exit.
“Ed Woodward told me, on the record, that last week was not a proud one for Manchester United,” White said.
“He said the club now needed to undergo the proper process of finding the right manager and that nothing will be announced before the end of the season.
“He said he’s been inundated with applications and recommendations – the top people in the game want this job.
“But he said, at the end of the day, nothing will be announced before the end of the season.”
Solskjaer could be among the contenders for the full-time job after an excellent start that has seen him win his first two games in charge.
The Norwegian has made no secret of his desire to stay beyond his initial six-month deal.
“I’m happy to help out and my job now for the next six months is to do as well as I can and move the club forward as well as I can,” he previously said.
“I understand there are so many managers who would love to be manager of Manchester United and I am one of them.
“But it is not something we’ve talked about, they’ll do a process now for the next six months.”
He was quizzed again own the subject ahead of their clash with Bournemouth at Old Trafford on Sunday.
”My job is for the next five months because I’m going to leave them to someone else when they come in,” he added.
“That’s the plan and that’s what my job is at the moment, it runs out at the end of May – actually the end of June but we don’t play many games in June – so my job is to affect the players on what it is all about at Manchester United.
“You’re always ambitious of course, but as I’ve told Ed [Woodward, United chief executive] and the owners, I’m here to do as well as I can and, if in May you decide someone else will be coming in, then fantastic, and if you decide it’s me, I’m sure we will agree.”