Transparency campaigners have sent a letter to the government, arguing that politicians and staff could avoid accountability by using these features.
Members of both the Cabinet and Shadow Cabinet use Whatsapp and Signal.
The Cabinet Office said records of official communications are "retained in line with guidance".
The legal challenge is brought by campaigning law group Foxglove on behalf of non-profit political campaigners the Citizens.
Its letter warns that if a satisfactory response is not received within 14 days, it may proceed to a judicial review.
The organisation has previously sent "several" FOI requests seeking disclosure regarding how the government uses instant messaging services.
It argues that the ability to remove or delete messages poses serious risks to transparency and democratic accountability.
According to official guidance, which has existed since 2013, a record only needs to be retained "if it is needed for substantive discussions or decisions in the course of conducting business".
For example, this law would apply to messages exchanged between special advisors and a minister regarding government policy.
UK law requires that such messages be archived to record, and it is up to the originator or recipient of such messages to "take the steps" to ensure this is done.
A spokesperson for the Citizens said there is a growing risk that the current period of British history could be little more than a "black hole".
'Cloak of secrecy'
"We are in an unprecedented national emergency and we are going to have no records of how decisions were made or even who made them," the spokesperson added.
"Government business is being conducted under a cloak of secrecy enabled by the tech platforms. The only way we can have any hope of holding power to account or even simply maintaining the historic record is through transparency."
A spokesman for the Cabinet Office told the BBC that "appropriate arrangements" are already in place to adhere to guidance, and "this is kept under periodic review."