"It doesn't matter how many games, I'll stand there a proud man and give it everything I've got," the 44-year-old Unsworth, who made 314 appearances for Everton, told a news conference ahead of Wednesday's League Cup trip to Chelsea.
The former defender said Koeman paid the price for failing to win - Everton have two league victories in nine games under the Dutchman this season.
"The bottom line for any manager when they lose their job is did you win enough games and unfortunately we're in a position where you're going to say 'no, we haven't'," Unsworth said.
Unsworth, who filled in as caretaker for the final game of the 2015-16 season after Roberto Martinez was sacked, said he had no qualms about taking the interim role for a second time.
"It's a very easy decision to make... it's not a difficult decision at all to stand as Everton manager," he added. "If you're an Evertonian, that makes you very proud.
"I'll repeat myself from the last time I sat here 18 months ago. Who wouldn't want this job?
"It is a top job for absolutely any football manager, myself included... an opportunity to hopefully get a run of games where performance levels go up and we produce some wins as well.
"From that, performances and results will dictate any future for me as Everton manager. I want to manage and this is an amazing club to be manager of. I think the time has come to say that. I make no secret of it."
Unsworth refused to single out individuals responsible for the team's slump.
"The problem is the team hasn't won enough games and we have to burden the responsibility together. It's not down to individuals leaving, individuals not coming in, it's a team game and we burden that together," he said.
Third-bottom Everton visit 14th-placed Leicester City, who last week sacked manager Craig Shakespeare, in the league on Sunday.
(Writing by Simon Jennings in Bengaluru, editing by Ed Osmond)
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