Keane has had a mixed managerial career to date, leading Sunderland into the Premier League at the first time of asking before leaving the Stadium of Light in late 2008.
He took over at Ipswich towards the end of last season and has spent this term waging a battle against relegation from the Championship, with recent reports suggesting time is running out for him at Portman Road.
And, rather than worry about succeeding Mowbray at Parkhead, Keane confirmed his priority is to improve Ipswich's fortunes after winning just 13 of his 46 matches at the helm.
"The way I've been doing my job at Ipswich I shouldn't be linked with any other jobs," Keane told Sky Sports News. "I need to focus on my own job and get results."
He added: "There's speculation about my job, but there are much better managers than me who have lost their jobs.
"If the club wants to move me on, then they'll move me on and there's very little I can do about it.
"This sort of talk doesn't keep me awake at night and my players shouldn't take any notice of it either.
"If you want me to comment on speculation then we'll be here all day and what good would that achieve.
"If I lose my job, I lose my job. There's always speculation about lots of managers in the Premier League and Championship so it's nothing new.
"If you think I'm fearful of that, don't kid yourself. I've got a good wife and five good kids so I'll be OK whatever happens."
Scottish Cup 2009/10 Win Outright: Celtic 2/5