Everton are still looking to find a permanent replacement for Ronald Koeman.
Unsworth is currently in temporary charge of Everton, but his future is not yet clear.
The Toffees under-23 boss has won just one of his four matches in charge since taking over from Ronald Koeman, but he has still impressed Everton’s board with his management skill.
Kenwright would now prefer to appoint Unsworth over Allardyce, but the former England boss does have his supporters at Goodison Park.
Allardyce is currently out of work, and his track record of keeping clubs in the Premier League could make him an appealing prospect for the Toffees.
Everton are currently in 15th place in the top flight standings, after what has been an extremely disappointing campaign to date.
Everton are expected to appoint a new manager in the coming days, with the Merseyside club’s next match coming against Crystal Palace after the international break.