Al Hasawi, who is also Forest chairman, has been on the hunt for a new boss after sacking Billy Davies earlier this week and has seen former QPR manager Neil Warnock and club legend Stuart Pearce both turn down the role after talks.
The Kuwaiti businessman has admitted difficulties with a number of his targets, with Clarke still on gardening leave after being sacked by West Brom, Mackay facing a court case with Cardiff owner Vincent Tan and Roberto Di Matteo, whom Al Hasawi ruled out of the running, still being restricted by his previous contract at Chelsea.
Speaking exclusively to Sky Sports News, Al Hasawi said: "Steve Clarke, Malky Mackay and Gianfranco Zola are on the list, yes, but there are things I would like to mention because I have not spoken to them, I have spoken with their agents.
"Every single manager we wanted to take they say he is on gardening leave or he has some problem with his former club, or they say 'we don't want to do it right now, we want to do it in the summer'.
"Many managers want to start at the beginning of the season - most of them do.
"They don't like to start for the last eight games. Zola is someone who wants to start in the summer.
"I don't want to say I am near to appointing someone right now because every second I have a phone call regarding a manager.
"You never know, maybe tonight we hear something positive and we go for it, but I don't want to rush.
"I want to pick the right manager. We know what the fans want and what they don't want and what will make them happy, and if we are going to pick anybody I am sure they will be happy."
Earlier, Al Hasawi claimed he was aiming to have a new manager in place by the end of next week.
It remains unclear whether Al Hasawi, assisted in his search by former European Cup-winning captain and recently-appointed club ambassador John McGovern, is looking for a permanent manager or someone to take charge for the remainder of the season but the Kuwaiti is hopeful of filing the role in some capacity within the next 10 days.
"Hopefully by the end of next week we will have a new manager in place. I am working closely with John McGovern on a daily basis," Al Hasawi said via his Twitter account.
"We don't want to rush into a decision. We have to get the right man and we are prepared to wait to achieve that.
"In the meantime we must all stay behind Gary Brazil. There are important games ahead and the play-offs are still within reach."
Former Forest captain Pearce turned down the chance to return to the club as manager on Thursday.
Pearce, who played for the club for 12 years, said earlier this week that he still had managerial aspirations and described the job as an "outstanding opportunity" for the right person.
However, the 51-year-old feels he is not that man after holding talks with Al Hasawi about succeeding Davies.
He told talkSPORT: "I've had a chat with them but at this moment in time, for personal reasons, I've decided not to take it any further."
Asked if he was tempted by the offer, Pearce added: "There was an emotional pull because I had 12 great years at that club."
Pearce would have been a popular choice among Forest fans.
The tough-tackling left-back spent the majority of his career at the City Ground where he twice won the League Cup under Brian Clough.
During his final season with the club in 1996/97 he also had a brief spell as caretaker manager.
It has been reported that interference in the signing of players and in team selection from Al Hasawi was a factor in both Pearce and Warnock turning down the job.
However, the Kuwaiti insists this is not the case.
"I would like to categorically state I have no desire to interfere in team affairs. The manager's job is to manage," he added.
Brazil, Forest's academy manager, is poised to take charge for the second successive game when Forest attempt to get their Sky Bet Championship promotion challenge back on track at Ipswich on Saturday.
Brazil's first game saw the Reds lose 1-0 at home to Charlton on Tuesday night - a result which left the club seventh in the table and two points shy of a play-off spot.