Bernstein ruled that Chelsea defender Terry should be stripped of the England captaincy after being charged with racially abusing Anton Ferdinand of Queens Park Rangers in October 2011.
Terry and Chelsea team-mates Frank Lampard, Branislav Ivanovic, Petr Cech and Fernando Torres were at the handover in London's Whitehall when the trophy they won last year was presented to Britain's Sports Minister Hugh Robertson by UEFA President Michel Platini.
The Chelsea skipper rejected Bernstein's attempt to shake hands when the two came together after the presentation ceremony.
"It's a difficult one for me - he went and spoke about me in the court case," Terry told Sky Sports television.
Asked if he shook Bernstein's hand, the Chelsea defender replied: "No".
Terry was cleared of the racism charge in court but the FA later banned him for four matches for the incident that occurred in a Premier League match between QPR and Chelsea at Loftus Road.
Bernstein said he had no idea the Chelsea player had snubbed him.
"My relationships with everyone here are really fantastic," the FA chairman told reporters. "I didn't notice a thing and I think we should talk about more positive things than that sort of nonsense."
Asked about his relationship with Terry, Bernstein said they were "a little distant" but he did not seem too bothered about improving matters.
"I don't know," he added. "I'm not really terribly concerned, I have other things to think about. In my last three months in the job I have other priorities."
This season's Champions League final is being staged at Wembley Stadium for the second time in three years to mark the 150th anniversary of the FA.
The May 25 showpiece will be London's seventh final following previous games in 1963, 1968, 1971, 1978, 1992 and 2011. No other city has staged the match as many times.
The women's Champions League final is being played at Chelsea's Stamford Bridge stadium on May 23.
Earlier on Friday the two trophies were carried around London on public transport to promote the events.
Champions League ambassador and former England and Chelsea defender Graeme Le Saux said: "Taking the trophies on the tube and bus was a great way to start the day and I'm sure it brightened up the morning for some of the commuters".
Next week Bayern Munich play Barcelona and Borussia Dortmund meet Real Madrid in the first leg of the semi-finals.
