Everton are 18th in the league and bottom of their Europa League group but Unsworth said victories over Olympique Lyon and Watford in their next two games would give the team a much-needed boost and do no harm to his hopes of being named manager.
"Do I need to win the next two games? Probably," Unsworth told reporters ahead of Thursday's match against Lyon. "You have to get results, so we need to start winning. But that doesn't scare me.
"I've not looked at the other names linked (for the managers position)... but whoever has been linked with the job, it won't surprise me because this is a wonderful club."
Defender Ashley Williams said the players were hurt by Koeman's dismissal and that they must share some of the blame.
"It's always disappointing when a manager loses his job. You do take responsibility for that," he said.
The Wales international added that Everton must start winning for them to build momentum.
"We need wins and we need them fast," he said.
"We'll try and win the game and after that we'll recover and we'll go against Watford and we'll try and win that.
"We'll continue to do that until we get a bit of momentum and confidence starts to build and then your season can start."
(Reporting by Aditi Prakash in Bengaluru; Editing by Peter Rutherford)
