The newspaper is preparing to stop using its official X accounts, having joined the social network in 2007, when it was known as Twitter.
Reporters are being told they can remain on the app with their personal profiles, multiple sources said. A formal announcement is expected soon.
The last post on The Guardian's main X account was a link to the story "Birdwatch: a close encounter with a flying rhino in Borneo", at 7.14am on Wednesday.
US news sites including NPR and PBS left X last year after they were labelled as state affiliated media under rules introduced by Mr Musk.
Thousands of social media users have quit X in recent days following Mr Trump's election win to join rival app Bluesky.
Bluesky, which was originally launched by Twitter founder Jack Dorsey as a research project, has added more than 1m users over the past week with the app closing in on 15m users.
Development started in 2019, while Mr Dorsey still ran Twitter, as a "decentralised" version of the app. It was spun-off after Mr Musk's takeover and officially launched last year.
The app is now primarily owned by Jay Graber, its chief executive, and operates as an independent company. In October it raised $15m (£12m) from investors Blockchain Capital.
Bluesky is currently ranking top in the US iPhone App Store, while it is fifth in the UK store.
Threads, a rival service with a similar design to X that is owned by Mark Zuckerberg's Meta, has swelled to 275m users since it launched in July last year.
The billionaire and world's richest man spent election night with Mr Trump in Mar-a-Lago and campaigned for the Republican for weeks. Mr Musk claimed last week that X had enjoyed "record usage" during the US election.
On Tuesday, Mr Trump confirmed the Tesla boss would take the helm of a new department of government efficiency to enact sweeping cost cuts in the US.
Mr Musk's X has endured an exodus of advertisers since his takeover in 2022 amid concerns over the social media site's light-touch moderation policies. Mr Musk has emphasised freedom of speech and reinstated thousands of accounts previously blocked from the site for breaking its rules.
A Guardian spokesman declined to comment. X was contacted to comment.