Unsworth announced on Tuesday that the talks are still ongoing between the club and a couple of people.
“As we sit here right now, I believe the board are talking to a few people.
“Talks are ongoing, but nothing has been finalised yet with anybody. So I continue to do what I was asked to do five weeks ago.”
Shaktar Donetsk coach Paulo Fonseca and Republic of Ireland boss Martin O’Neill have been listed as candidates.
Unsworth stepped up from the Under-23s following the sacking of Koeman on October 23.
The caretaker also said that he might still be in charge for Wednesday’s clash with West Ham.
When asked about Allardyce, Unsworth said: “Sam is a top manager, but I think it would be wrong of me to talk about individuals.
“I have seen the names that have been bandied about – all top managers. Whether there’s truth in the names of some, or all, or none, I don’t know.
“But whoever gets the honour of being Everton manager will have my full support and my full backing.”
David Unsworth only won one of the seven games after being the caretaker of the Toffees. He also showed interest in becoming Everton’s boss but this would be unlikely as Everton stay in relegation.
“I think we need our best players fit, quickly, some proven Premier League recruitment in early January and I think we need to win a couple of games and get confidence flying again.
“When confidence is low, it takes a bit of time to build that back up. I don’t think there’s a quick fix – I think whoever gets the job needs time, patience, support and backing.”