More than 95,000 people have signed the online petition urging the UK government to hold its own, independent investigation into the Thai case.
The Thai police investigation has been widely criticised, including by the UK.
The bodies of the British tourists were found on a beach in Koh Tao last month. Two Burmese men are under arrest.
The petition is due to be handed into Downing Street later by Burmese activists who also plan to hold a demonstration in Whitehall.
Post-mortem examinations found Mr Miller, 24, from Jersey, died from drowning and a blow to the head, while Miss Witheridge, 23, from Hemsby in Norfolk, died from head wounds.
Zaw Lin and Win Zaw Htun, both migrant workers from Myanmar, also known as Burma, are suspects in the murders.
They are charged with conspiracy to murder, conspiracy to rape and robbery, and could face the death penalty if found guilty.
Police have said DNA found on Ms Witheridge matched samples taken from the men.
But concerns about the murder investigation include the fact the crime scene was not sealed off after the killings, and an early statement by police that no Thai person could have committed such a crime.
Mr Zaw's mother said this week her son was being made a "scapegoat" by police.
Phyu Shwe Nu told the BBC "the case has been fixed", while both suspects' parents said their sons were innocent.
BBC political reporter Chris Rayner said the reports have pushed up the number of signatures on the petition.
On Monday, the UK Foreign Office summoned the Thai charge d'affaires to express "concern" how the investigation had been handled.
A petition calling for an independent investigation into the murders of Hannah Witheridge and David Miller is to be handed in at No 10 later.