If the petition which is on website change.org breaches the 100,000 mark, the union plans to deliver it to 10 Downing Street and push for a meeting at Defra. The petition reached 5,000 signatures in its first two days.
Flanagan said: "We see that figure as quite significant and believe that the Government and Defra will see some significance in getting to that level."
Flanagan said that a Parliamentary drop-in session about staff concerns held in the House of Commons on 14 May was "very successful".
An Early Day Motion calling for a public debate on the cuts, tabled on April 9, has now been signed by 34 MPs from across the political spectrum. Prospect and GMB union representatives also attended RHS Chelsea Flower Show last week to drum up support for their campaign.
Paterson has told Horticulture Week that an emergency payment to bail out Kew was out of the question.
Speaking to HW during Chelsea week he said the Government is spending more on Kew now than during the previous spending round and insisted that the 125 redundancies at Kew were not his responsibility.
Ann Ainsworth, officer at union PCS, said that Paterson won his Shropshire seat on a majority of 26,692 votes in 2010. "If that is sufficient basis for him to become an MP and then selected to be Secretary for the Department of the Environment Food and Rural Affairs then four times that number of people showing
their support for Kew, not just in the UK but around the world, ought to make him listen."
Kew is facing a £5 million funding shortfall with current director Richard Deverell planning to find the money through a combination of increased income generation and job cuts - with as many as 125 of the current 750 jobs under threat.
Most are in the sciences. Flanagan said staff at Kew remained very nervous as they await the results of a restructuring process due in July.
The news of the cuts has been greeted with dismay across disciplines with high profile figures including broadcaster and naturalist Sir David Attenborough, former Regius Keeper of Royal Botanic Gardens Edinburgh Stephen Blackmore and Richmond Park MP Zac Goldsmith criticising the decision to downgrade services.