Unsworth is in charge of Everton at the moment, having been placed in the role following the departure of Ronald Koeman.
According to BBC Sport, Everton are interested in appointing Marco Silva as their new manager.
Silva was appointed the Watford manager in the summer of 2017 following his departure from Hull City at the end of last season.
Former Arsenal star Merson has suggested that Everton should hand the managerial role to Unsworth until the end of the season.
Merson wrote in The Daily Star: “There are too many games coming up in quick succession. A new man won’t have the time to put his stamp on it or work on anything in training.
“Caretaker David Unsworth should get it until the end of the season and then they can look at it again.
“Unsworth has just masterminded a huge comeback win over Watford. That doesn’t happen if Ronald Koeman is still in charge.”
Everton are struggling at the moment and are in danger of getting relegated to the Championship at the end of the season.
The Toffees need to decide on a new manager soon, and if they think that Unsworth is the right man, then they should hand him the role until the end of the season.