More than 120,000 of the names have been collected by Exeter postman Darren Rowbotham as part of a growing campaign against the sell-off.
Union leaders, Labour MPs and campaigners will join a delegation urging ministers to reconsider the privatisation.
The move follows an announcement by the Communication Workers Union of a strike ballot among 125,000 of its members over issues linked to privatisation.
The ballot is due to end on October 3, with any walkout starting from a week later if there is a yes vote, threatening action in the run-up to the busy Christmas period.
CWU members voted in a recent consultative ballot by 96% against privatisation, with a similar number saying they would be prepared to boycott delivering competitors' mail.
The union warned a strike was "inevitable" unless a deal was agreed on a number of issues, including pay, further changes to workers' pension scheme, the impact of possible privatisation on job security and terms and conditions, and the company's future strategy.
The Royal Mail has warned that any industrial action, or the possibility of disruption, was damaging to the business, especially in the run-up to Christmas.
Meanwhile, the Business Select Committee has postponed a one-off evidence session on the Royal Mail with Business Secretary Vince Cable, originally scheduled for next Tuesday, until a date to be confirmed in October.
A Department for Business spokesman said: "Action taken by the CWU will not alter the Government's decision to sell shares in Royal Mail in this financial year."