The agreement in principle came following the recommendations of the Migration Advisory Committee (MAC), which published a Government-commissioned report last week.
The study said that, if immigration is not part of the negotiations with the EU and the UK is deciding its future system in isolation, there should be no preference given to citizens from the European Economic Area (EEA), which includes the present 28 EU countries, as well as Iceland, Norway and Lichtenstein.
READ: Government documents reveal potential no-deal Brexit consequencesAt a Cabinet meeting on Monday, ministers reportedly backed a system which would be based on skills rather than nationality following a presentation by MAC chairman Professor Alan Manning.
A Downing Street spokeswoman said: "Cabinet was told by Professor Manning that the MAC was recommending that, in the post-Brexit immigration system, EEA and non-EEA nationals should be part of one universal system.
"A key conclusion was that high skilled migration is of greater economic benefit than lower skilled migration, and as such the MAC recommended that the new system should make it easier for higher skilled than lower skilled workers to come to the UK.
"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."
The MAC review published last week recommended ministers scrap an annual cap of 20,700 on the number of visas available under the Tier 2 skilled work scheme, and open up the route to "medium-skilled" jobs.
READ: The Yorkshire Post says: The three-way Brexit battle. Deep divisions reveal Theresa May’s taskBut in a finding it acknowledged would attract opposition from some sectors, the MAC concluded that there was no need for a specific migration route for low-skilled work, with the possible exception of a seasonal agricultural scheme.
It said this would not mean there is no supply of low-skilled migrant workers, stating that most of the existing stock would remain and there would likely be a continued flow through family migration or the existing youth mobility scheme.
Yesterday's Cabinet meeting also saw Theresa May tell ministers to hold their nerve as Brexit negotiations hit a difficult stage in the wake of the rejection of her plans by EU leaders at Salzburg.
The Prime Minister said she remained confident of securing a withdrawal agreement with the EU, but the Government would continue to plan for the possibility of no deal.