It would be good to have a little more information on him.
Please think about making a wikitree profile for him.
With what information I have the only one I can see is in the 1871 census
There is one David Osborne born to William Osborne & Jane in Liverpool, but all three were born in Belfast Ireland
David is 6 years old and his parents 26
Ref Piece: 3758 Folio: 105 Page Number: 42
If David grew up in Liverpool he may have just excepted he was born there, without ever being told that was the case.
Hope this helps