"I have got to win as many games as I can and get as close to Rangers as we possibly can in the league and try to win the cup," he said.
"You just take the job and see where it takes you, even if things go well it may not be offered to me."
Mowbray was axed following the 4-0 drubbing by relegation-threatened St Mirren in midweek.
And Lennon, who took his first training session with the squad on Friday morning, is demanding an improvement.
"Up until the St Mirren game things were going okay, but the players understand the performance was unacceptable," he said.
"I have made it clear what is expected of them between now and the end of the season.
"I would like us to be more direct, press teams more and get the crowd going; speed up the tempo of the game.
"They have trained well this morning and we are looking forward to tomorrow's game and putting things right."
I never really buy into the bookmakers' favourite; it doesn't wash with me
Norwich City manager Paul Lambert
Former Celtic captain Tom Boyd told BBC Scotland that the club should recruit an experienced manager to succeed Mowbray.
Boyd conceded that Lennon could force himself into the reckoning with a winning run, but he expects the club to go for "someone who's been over the course before".
'I'm not sure if Mark Hughes is experienced enough at this kind of level. I know he's been down in England in the Premier League and did well with Blackburn," Boyd said.
"He did okay with Man City but he had plenty of money to spend there. I certainly don't think he'd get that kind of money if he came up here.
"I'm certainly looking forward to seeing who is going to come in and I don't think the board will be in too much of a rush."
Norwich City manager Paul Lambert cooled speculation linking him with a return to his former club.
"I can't stop people from saying things but I love it here," said Lambert.
"It's been great. As I said before, Celtic are a terrific football club and I had eight great years there.
"But I'm happy here, that's for sure. I never really buy into the bookmakers' favourite; it doesn't wash with me. I just do my job."
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