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.
The petition, launched earlier this month on the website change.org, by London resident Julie Flanagan reached 5,000 signatures in its first two days.
In her most recent update on change.org on April 16 she said: "Enormous thanks for all the amazing support that has been given to this campaign" giving a range of suggestions for ways which supporters can protest against the cuts.
She has called on constituent supporters to lobby Richmond Park MP Zac Goldsmith at his next local surgery on Friday. Goldsmith has come out in support of the campaign, according to a story in the Independent. He called Kew "a national jewel" and said it would be a tragedy if its work and reputation were to suffer.
An Early Day Motion tabled on April 9 calling for "a full public debate has so far attracted nine signatures from MPs. Flanagan is also encouraging supporters to write to their MPs and urge them to sign too.
EDMs are one of the few ways backbench MPs can get a subject debated in the House of Commons, although few are given the chance to do so. However they do raise awareness.
The news of the cuts has been greeted with dismay across disciplines with high profile figures criticising the decision to downgrade services.
Sir David Attenborough told local newspaper The Richmond and Twickenham Times that Kew played "an absolutely crucial role in looking after out botanical heritage and our botanical future"
He added: "People who think it is just a place to go to look at pretty flowers and flower beds are mistaking the importance of Kew Gardens."
Former Regius Keeper of Royal Botanic Gardens Edinburgh Stephen Blackmore also called on the Government to rethink its plans, saying that Kew was a vital bit of world heritage."
He told Horticulture Week there was "a sense of shock inside Kew".
TV ethnobotanist and author James Wong has also been vocal in his anger about the cuts. Wong, who trained at Kew and also worked there, has been rallying support for the petition on Twitter.