According to a 'Sunday Times' report, the British government is planning to pilot a scheme from November targeted at visitors from at least six countries, also including Bangladesh, Sri Lanka and Ghana.
Tourists aged 18 and over would be forced to hand over 3,000 pounds for a six-month visit visa, which they will forfeit if they overstay in Britain.
Initially the scheme will target hundreds of visitors, but the plan is to extend it to several thousand, according to the front-page report in the newspaper.
Cameron's election pledge was to cut annual net migration below 100,000 by 2015.
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"This is the next step in making sure our immigration system is more selective, bringing down net migration from the hundreds of thousands to the tens of thousands while still welcoming the brightest and the best to Britain," May has been quoted as saying.
"A Home Office official said the six countries highlighted were those with "the most significant risk of abuse".
Last year, 2,96,000 people granted six-month visas were from India, 101,000 from Nigeria, 53,000 from Pakistan and 14,000 each were from Sri Lanka and Bangladesh.
Several hundred visitors will be selected from each country for the trial scheme.
But ministers plan to extend it to all visa types, including work and student visas, and to all countries, although it will not cover every one of the 2.2 million people granted visas each year.
A second category will cover lower risk countries such as Kenya, where British officials believe the applicant to be genuine but have "residual doubts" about their intentions to return home.
The introduction of these bonds is expected to face criticism from pro-immigration groups because it does not target groups from the so-called "white Commonwealth".
A similar scheme has already been rejected in Canada amid claims that it was unlawful because it discriminated against categories of immigrants.