President Reagan and Andrei Gromyko had been talking together for quite a while. Again and again, Mr. Reagan probed for a way to soften his companion's unbending position on finding a way to improve US-Soviet relations. The President's shoulders slumped. He seemed to be giving up.
Then, putting aside the formal approach for the moment, the President said something like this: ''Look, I'm 73 and you're 75. We are a couple of old men who want to leave a peaceful world behind. Why can't the two of us do something right here that will put our nations back on the road to peace again?'' At this point, Gromyko rose, ending the conversation. But as he did so, he smiled and said, in a friendly way, ''We will talk.''
Now this is the report from an administration official who observed this incident. It may bear a pro-Reagan slant. But here is an anecdote that the President might well relate in his next debate with Walter Mondale. What a lift it would give his final summary, particularly if it is part of a Reagan pledge to give the highest priority to a search for peace during a second term.
Two of the President's closest White House associates have assured this reporter that Mr. Reagan wants, above everything else, to move the world closer to peace in the next four years. They say that shoring up the economy remains a prime presidential objective - including efforts to cut spending and bring about a tax reform that will end many inequities.
But they say Mr. Reagan wants most of all to be remembered as a President who was a peacemaker.
They say, moreover, that he would love to confound those critics who say he has brought back the cold war, or something close to it, through the use of cold-war rhetoric in the early part of his administration.
The point, however, is this: In his first debate the President - too much, anyway - was put on the defensive by Mr. Mondale. Too often he was placed in the position of having to defend his record. This reflects the challenger's advantage.
But a president has his own terrain where, if he sticks to it as much as possible, he can look very good in a debate. That's when he is being presidential.
When he speaks of his priorities for a second term, he is being presidential. And if he can provide some insight into his vision for America for the years to come, he can be very presidential.
But spending most of his time defending his record and answering the challenger's criticism - that's not being presidential. It is playing into his opponent's hand.
Mondale can and does talk about what he would do as president. But his words don't carry the weight of those of the man who has carried full White House responsibilities for nearly four years - and who, clearly, is still running the nation.
Quite obviously, Mondale knows how to make the most of his time when in the ring with Reagan. He has mastered the art of showing respect and warmth for the President while, at the same time, swinging hard at what he alleges to be the Reagan record. He's out to unhorse the President, to bring him down to his own level. And he did a pretty good job of this in the first debate.
Who won the debate? My own scorecard had the two running about even. I wasn't surprised, however, when the viewers gave Mondale the edge - even though the Minnesotan had not persuaded many Reagan supporters to shift their votes.
The perception one got on TV was of a Reagan who was overstuffed with facts and, in the last half hour, less composed than usual. There was a faint reminder here of the 1960 debate when Richard Nixon, looking tired and grim, lost the ''perception'' debate to a poised challenger, John Kennedy. Mr. Nixon was adjudged by many observers to be the winner on points - and this edge came through clearly on radio, where debating substance and the arguments of the two men were the decisive factors.
Reagan, if he can be the familiar Reagan, witty and in command, can win the ''perception'' debate next time - but only if he stays on his own terrain and avoids getting dragged into a defensive pose where he doesn't look very presidential and thus where he becomes vulnerable.