Moshiri told Talksport that the club were close to securing a replacement for Dutchman Ronald Koeman who was sacked on Oct. 23 after the club dropped into the relegation zone.
"The first priority is to stabilise the club and finalise the appointment of a manager. We are close," Moshiri, a British businessman, said.
Former Everton player Unsworth was elevated from his role as under-23 manager and has overseen one victory and four losses in six matches including the 5-1 hammering at Atalanta in the Europa League on Thursday.
"If we're further than we were last night, or before we went on to the training pitch, towards getting a permanent manager that's fantastic," Unsworth told a news conference on Friday, ahead of Sunday's Premier League trip to Southampton.
"The sooner it's sorted the better, I think that would be fair comment. What will be, will be. This is a wonderful job for any aspiring manager, any senior manager, any manager of any club in the world."
Unsworth said Everton "owed their fans a win" after a poor performance against the Italian side but added that they could expect another tough game at Southampton who are winless in their last three games.
"I am really looking forward to it. It will be a tough game. Southampton are a good team, they pass the ball really well. But we must see a reaction now. Our players must stand up and have the courage to go and get a result on Sunday."
Everton are two spots and three points above the relegation zone after three wins, three draws and six defeats in their 12 games.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Ian Chadband)
