It's not in the movies and it wasn't a one-to-one chat but Dumbledore and Malfoy do have one private(ish) conversation before the end of Half-Blood Prince. When Malfoy, Crabbe and Goyle dress up as Dementors to try and distract Harry during a Quidditch match McGonagall says that she's going to talk to Dumbledore about the incident.
"An unworthy trick," she was shouting. "A low and cowardly attempt to sabotage the Gryffindor Seeker! Detention for all of you, and fifty points from Slytherin! I shall be speaking to Professor Dumbledore about this, make no mistake! Ah, here he comes now!"
(Prisoner of Azkaban, Chapter 13, Gryffindor Versus Ravenclaw).
The conversation takes place off-screen but, since Dumbledore appeared at that very moment, it seems likely that McGonagall would've quickly collared him and that he would've dealt with the situation immediately. Malfoy, Crabbe, Goyle, McGonagall and Dumbledore therefore all took part in a conversation where the latter possibly handed out some form of (additional) punishment or words of discipline to the three miscreants. This is the only personal conversation that Malfoy has with Dumbledore until they meet at the lightning-struck tower.