Everton are looking for a new manager following the recent departure of Ronald Koeman.
Unsworth has been in charge of Everton for the past two competitive games - against Chelsea in the fourth round of the EFL Cup and against Leicester City in the Premier League.
Former Arsenal striker Wright has written in The Sun that the 44-year-old former Everton defender is not the right manager for the Merseyside outfit.
“Who knows, there’s no reason why he can’t eventually do it at the club he loves and where he is currently steering the ship,” Wright wrote about Unsworth in The Sun.
“But if Everton hand him the reins now ahead of time, it could turn out to be the worst move either of them could make."
“If anyone truly believes doing a great job with the Under-23s qualifies you as a Prem manager, they’re sadly mistaken,” added Wright.
"The Toffees have developed a reputation as one of the country’s top breeding grounds for young talent, and Unsworth has played a massive role in that.
“But that doesn’t mean he is ready to take first-team control now. I think we got a glimpse of that at Leicester on Sunday.”
Everton are third from bottom in the Premier League table at the moment with just eight points from 10 matches.
The Toffees have a squad good enough to push for a top-10 finish in the league, but the players have not performed to the level they were expected to at the start of the season.