Posting on Twitter, he said successful candidates would then need 32 in the second ballot to remain in the contest.
The 313 Conservative MPs will continue to cast their vote and whittle down the Tory MPs until there is only two candidates remaining.
Mr Fabricant concluded: ''It is hoped the final 2 will be chosen by the end of next week and a new PM will be in place by the end of July at the latest.''
Over 124,000 Conservative party members will then vote on the final two.
Theresa May will quit as Tory leader on Friday June 7 and Conservative parliamentarians have until the week commencing June 10 to put their name in the ring.
Boris Johnson, Michael Gove, Rory Stewart and Esther McVey are just some of the candidates vying to succeed Theresa May.
Last week Nigel Evans, an executive member of the 1922 told ITV News the leadership contest was in danger of getting out of hand.
He said that with days still to go before declarations close, the field could reach up to 20 - and some candidates were or would be "clearly positioning themselves for a Cabinet job", rather than hoping to be the next leader, and that was wrong.
The Ribble Valley MP continued that if someone is putting themselves forward as leader of the party - and ultimately prime minister - then they should be able to display a degree of public support before entering the ballot.
"I think it is important that we have only credible people going forward, otherwise we will end up with 20-odd people, some of whom have absolutely no chance at all but what they do really want is a Cabinet job...
"Well, I've got a good idea, just support the person who you think is going to give you that and ideally support somebody who you believe in their principles," he said.
Former Tory leader Iain Duncan-Smith echoed the concerns raised about the size of the field, likening it to a "Charge of the Light Brigade".
He urged the Committee to look at the rules around who could run and how much support they actually had - even though, he suspected, a number of hopefuls would swiftly realise they had no chance of getting into Number 10.