Bernstein shook hands with members of a group of Chelsea players at the handover ceremony in the Banqueting House on Whitehall in London, including Bratislav Ivanovic, Fernando Torres and Petr Cech, but Terry kept his hands firmly by his sides.
The Chelsea captain later confirmed that he had rejected Bernstein's attempt to shake hands.
Bernstein was the man who stripped Terry of the England captaincy after he was charged with racially abusing QPR's Anton Ferdinand.
Terry was cleared at Westminster Magistrates Court of making a racist insult to Ferdinand, but was charged by the FA and banned for four matches.
Terry remains unhappy with Bernstein's role in the case.
Terry told Sky Sports News: "Listen, it's a difficult one for me. Obviously he was the one who spoke about me in the court case and said things. I don't want to talk on air. It's probably a subject that we should maybe just avoid."
Asked if he shook Bernstein's hand, Terry replied: "No."
Bernstein insisted he was unaware of the Chelsea player's snub when quizzed on the matter by Sky Sports News.
"I didn't notice anything," said Bernstein, who will stand down from his role in July. "My relationships with everyone here are really fantastic. I didn't notice a thing and I think we should talk about more positive things than that sort of nonsense."