Three Conservative MPs have been eliminated from the race after not gaining enough support.
There will be more secret ballots next week to whittle the remaining contenders down to a final two.
Former Foreign Secretary Boris Johnson received 114 votes in the secret ballot - far ahead of the runner-up Jeremy Hunt.
UK correspondent Olly Barratt says it cements Mr Johnson's role as favourite.
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UK correspondent Olly Barratt says Mr Johnson is likely to play it safe ahead of the next votes.
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