Moyes was left stunned at the news and reacted by telling his fellow Scot he could do some coaching at Finch Farm in the interim.
"I was with Alan last night and I have said to him he is welcome to come in. He is a student of the game and won't sit at home," said Moyes, who learned his trade at Preston.
"I was really disappointed for Alan. He was my assistant for a long time. I am also friendly with the people at Preston as well.
"So it was a shock to me as I think it was to everyone in football. They have a tight budget but every club is the same.
"Alan has done such a good job he will undoubtedly work again. It would not surprise me if he gets one of the big jobs going around at the moment."
Bolton are one club in the market for a new manager after sacking Gary Megson with Moyes appalled by the abuse the former West Brom manager received from supporters in the 2-2 draw with Hull after taking in the game.
The 46-year-old added: "I couldn't believe the treatment Gary got from the crowd. It was terrible. He didn't deserve it. His team were winning.
"The game was tight and he made a tactical change. The crowd affected the Bolton team which in turn didn't help them and they ended with a draw."
Moyes believes he is in a privileged position at Goodison Park as his team prepare to face League One Carlisle in the FA Cup looking to go one better than last season when they were losing finalists to Chelsea.
He continued: "I think I am at a really good club. You look at the way some managers are getting treated at present.
"I am really fortunate I am working with good people. Sometimes we have to be thankful for that. At Everton there has been a level of success because of good stability.
"We are a getting a big closer to winning trophies. We had a great run in the cup last season.
"Playing on the last day at Wembley was special and has given me a bigger hunger to do it again with a different result. That's why we are treating the competition seriously."
Moyes, however, will be short of strikers for the home third round tie.
Jo has been suspended by the club for returning to Brazil without permission, Aiyegbeni Yakubu is away with Nigeria while Louis Saha has a hamstring injury.
It means James Vaughan is likely to start and remain part of the squad, instead of returning to Derby on loan.
Meanwhile United States midfielder Landon Donovan will check in on Saturday to start his loan deal from Los Angeles Galaxy.
Moyes added: "I am looking forward to him arriving. He will be with us for two and a half months.
"Landon has a bit of pace, is a goalscorer and is experienced. We will be looking for him to get a good week's training in first of all. We know he is a very fit lad."