Ministers gathering on Monday “unanimously supported a system based on skills rather than nationality”, an unnamed source told the BBC, with similar reports in The Times and The Guardian.
Citizens from the other 27 European Union member states are currently free to live and work in Britain under the bloc’s freedom of movement rules, but this will end after Brexit.
A government-commissioned report last week recommended that in future, EU and non-EU citizens follow the same immigration rules, with preference given to high-skilled workers.
The report’s author, Alan Manning of the Migration Advisory Committee, briefed the cabinet meeting on Monday on his plan.
A Downing Street spokeswoman said: “The cabinet agreed that, once free movement is brought to an end, the government will be able to introduce a new system which works in the best interests of the UK, including by helping to boost productivity.”
May is now expected to make an announcement on future immigration rules at her Conservative party’s conference next week, before likely introducing proposals later this year.
However, any post-Brexit immigration policy could be affected by Britain’s future trade deals.
The EU may well seek to negotiate continued preferential treatment in return for access to its single market, while other countries could also seek visa waivers in return for trade deals.
May promised last week that even if Brexit negotiations with the EU break down, the rights of Europeans currently living in Britain “will be protected”.
Her spokesman said there would be formal proposals published “shortly”.