Although (as the other answer says) Harry does not suspect he contains a Horcrux, he does stumbles upon the truth accidentally and indirectly when he and Dumbledore first confer on the quantity of Horcruxes (having just heard the unabridged Slughorn memory), in Half-Blood Prince, Chapter 23 (at p.470 of the 1st edition as published by Bloomsbury):
‘He made seven Horcruxes?’ said Harry, horror-struck, while several
of the portraits on the walls made similar noises of shock and
outrage. ‘But they could be anywhere in the world -- hidden -- buried
or invisible --’
‘I am glad to see you appreciate the magnitude of the
problem,’ said Dumbledore calmly, ‘But firstly, no, Harry, not seven
Horcruxes: six. The seventh part of his soul, however maimed, resides
inside his regenerated body. [...]’
At this juncture, Dumbledore had only just learned that Tom Riddle (later known as Voldemort) was interested in the number seven, but he had been aware of Riddle's interest in Horcruxes for some time. Given, moreover, that Dumbledore had been aware of Harry's unique connection to Voldemort and Parseltongue fluency for over a year, it stands to reason that Dumbledore probably suspected the existence of a Voldemort Horcrux within Harry (these attributes are adduced by Dumbledore when he tells Snape about the fragment of Voldemort's soul in Harry, which Harry sees in the pensieve in Deathly Hallows shortly after Snape dies).
So, perhaps Dumbledore realised that Harry was actually right to say there were seven Horcruxes (and thus Voldemort's soul had been split in eight), but Harry's answer was correct for the wrong reason (Harry had failed to count the original body as containing a (non-Horcrux) part of the soul, but this error was cancelled-out by his ignorance of the extra Horcrux), hence Harry was none the wiser.
Evidently, Dumbledore preferred to withhold the information that Harry contained an extra Horcrux that was unintended and unknown to Voldemort. In other words, this 'correction' is actually an instance of prophetic irony (albeit one so artfully dissimulated that not many readers will remember it... for another example of this device but within a single book, see Eoin Colfer's Artemis Fowl and the Opal Deception, on the topic of Kraken numbers), flagging to the astute reader that there is a lacuna in the information Dumbledore imparts to Harry (at a juncture when Dumbledore has an established reputation for secrecy and imparting only partial information).