The move would see Sacarens RFC become the first major rugby club to play on artificial turf, if the £10m redevelopment project gets the go ahead.
Chairman Nigel Wray said: "There is a unique opportunity to create a fabulous stadium.
"Shaftesbury Barnet Harriers would thrive, Sacarens would use the artificial turf pitch around 16 times per year, and the rest of the time – up to 350 days per year – the field could be used by schools, local clubs and the wider community."
He continued: "Barnet Copthall would also become a new bug for the award-winning Sacarens Foundation, which has been working with more than 500,000 young people in London over the past decade, improving lives by running various courses that link education and sport."
Shaftesbury Barnet Harriers treasurer Geoff Morphitis commented: "We look forward to working with Sacarens to ensure the delivery of an outstanding sporting venue and to expand our work in the local community."