May is currently leading the race, followed by energy minister Andrea Leadsom who received the backing of former London mayor Boris Johnson on Monday.
UK justice secretary Michael Gove, work and pensions secretary Stephen Crabb and former defence minister Liam Fox are also in the running to replace Cameron following last month's referendum in favour of Brexit.
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The ruling Conservative party's members will choose from two candidates backed by most MPs and the winner is due to be named on September 9.
The Tory party's 330 MPs will vote in the first round of the election on Tuesday, when the poorest performing candidate is set to be eliminated.
Further rounds will take place on Thursday and the following Tuesday until the two final candidates remain.
May, considered the strongest candidate, currently has the public support of over 115 MPs, with Leadsom at 40, Gove and Crabb around 25 each and Liam Fox fewer than 10.
The candidates on Monday had the chance to make their case to be the next Conservative leader during a parliamentary hustings.
May, who had campaigned to stay in the EU, emerged in the lead among her colleagues with support from 35% of the parliamentary Conservative Party.
A Times/YouGov survey on Tuesday also put her ahead in the race, suggesting she would triumph by a 32-point margin if she went head-to-head with Leadsom in the final round.
According to the survey, the tough-talking politician is favoured as the strongest leader (63%), the one able to make the toughest decisions (58%) and also the unifying candidate in this vote (61%).
In reference to Brexit, 44% of those surveyed said they thought May would get the best deal from the EU, compared to a quarter who sided with Leadsom and 16% with Michael Gove.
The bookmakers also predict that it will be May who is most likely to be the next British prime minister.
However, with the bookies forecasting a remain win in the referendum on June, their predictions are not likely to be seen as very reliable this time.
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