Horticulture Week reported the impending move in July and now staff and residents say owners the Princes Trust are closing in on a deal, which London garden centres who were initially sounded out, say they were priced out of.
Fulham resident Tessa Darling said: "Local residents are upset that the garden centre will become a pet supermarket." She claimed the plantarea will become a car park.
Darling said the existing buildings were protected but that she hoped that adverse publicity would force the Trust to change its mind.
She added: "The area is very beautiful and a pet supermarket belongs on a trading estate rather than a residential part of London. People have written to the Prince of Wales protested. He’s a conservationist so how can he allow this to go through?"
Staff spoken to by Horticulture Week indicated it was likely Pets at Home would take over, but that contracts had yet to be signed.
A The Prince’s Trust representative said: "We are going through a rigorous tender process to select the most suitable buyer, which is being managed independently by surveying company, Colliers International.
"We are continuing to look at all options for future of the site, which is likely to be taken on by another tenant. As a charity, The Prince’s Trust is required to focus on its core activity of supporting disadvantaged young people into jobs and training. With this in mind, we are currently looking at selling the lease of the garden centre in order to raise funds to invest in support of more young people."
Princes Trust chief financial officer Amy Stirling, who was appointed in February 2014, is believed to have scrapped the previous tender process, which had reached the shortlist stage, and started again with more than just garden centre owners being targeted.
The Trust bought two-thirds of the lease for Fulham Palace Garden Centre from the London Borough of Hammersmith & Fulham for more than £500,000 last year.
The charity took over from previous owner Fairbridge Trust in 2011 but said because its "main focus is supporting disadvantaged young people into jobs" it is "looking at all options for the future of the garden centre", which it sees as non-core in its operations.
The centre originally helped disadvantaged people to get back into work but has recently been used as a fundraiser for the Trust, which helps disadvantaged young people.