The attacking midfielder’s mark of 107 goals has stood for more than four decades, but is coming under threat from United’s No 9.
Atkinson was at the Kassam Stadium last weekend to see Constable, nicknamed ‘Beano’, notch his 93rd goal for the club with a thumping finish in the 3-0 win over Rochdale.
The 69-year-old admitted it was only recently he became aware of his record, but predicted it would not be long before it was surpassed.
He said: “I was surprised when somebody mentioned it to me a few years ago, because I didn’t realise I had it.
“I never counted it or thought about how many goals I’d scored, that never came into it, but it’s a nice thing to have.
“James should do it, I would have thought he will.
“It’s lasted a while. I hope he does it quickly so I’ll be able to see it, because I’m getting on a bit now!”
Constable’s strike last weekend was his 13th of the campaign and with just Saturday’s trip to Accrington Stanley remaining, the 28-year-old looks set to finish the season as United’s top scorer for the fifth successive year.
But with 15 goals still needed to break the record, he will need to be in good form next season to reach top spot before his contract expires in 12 months’ time.
Atkinson attended the Roch-dale game along with a host of players, including his brother Ron, from the 1962/63 squad.
They were special guests to celebrate 50 years since the U’s played their first Football League campaign.
Atkinson said: “It’s always good fun when we meet up. Everyone still has the same sense of humour and we take the mickey out of each other.
“I don’t watch much football now. I was never a good spectator, but I always keep an eye out for Oxford’s results.”