The Mirror says Kane has been capped by England at Under-21 level and has made clear his ambition to break into the senior squad.
But O'Neill believes he can convince Kane - eligible through his Galway-born father - to spearhead the Republic's Euro 2016 qualifying campaign.
The Ireland boss was at White Hart Lane on Sunday to watch Kane as his old club Aston Villa were beaten easily.
O'Neill is also interested in Kane's team-mate Kyle Naughton, even though he too played for England U-21s.
The full-back qualifies for Ireland through his grandparents.
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