The health secretary is said to have aligned with transport secretary Grant Shapps and conservative backbenchers concerned about the requirement.
Javid, Shapps and other senior ministers are expected to sign off on the decision to scrap the documents at a meeting of the Covid operations committee on Monday (14 March).
The Guardian, quoting a government source, reported the "expectation is to remove all remaining Covid border measures" and speculated the requirement for unvaccinated travellers to get tested on arrival could also be dropped.