Alberto Costa told ITV News that a future leader should consult MPs via their whips about their top team - to ensure people were selected on merit, and not because of their abilities in leadership campaigns or with "spreadsheets".
Mr Costa argued that what may have gone wrong under Johnson was a failure of good governance due to a failure of Cabinet government.
"Because some members of the Cabinet appear to have been tripping over themselves to pay homage to Boris Johnson rather than being a constitutional check on the prime minister, which is one of the principal purposes of Cabinet government under the British constitution," he told ITV News.
Others have criticised the outgoing prime minister for surrounding himself with yes-people, instead of drawing on a wide web of talent across the party.
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The MP - who sits on the committee that will judge whether Mr Johnson misled parliament over partygate - called on all the candidates for leadership to show evidence that driving up standards is "at the core of their work".
He said to do that they need proper Cabinet government.
"This can be done by ensuring the best candidates are chosen on merit and not simply because of their skill set in leadership campaigns and spreadsheets," Mr Costa added
"Accordingly backbench colleagues - through their whips- should be consulted on the members of the proposed cabinet members to ensure a majority of parliamentary colleagues approve the next leader’s choice of cabinet members."
He admitted that selecting candidates via a vote of the parliamentary party was another way to give MPs more control.