Mr Turnbull has set a deadline of midday tomorrow [[Friday]] for Peter Dutton to obtain the signatures.
The A-B-C is reporting 40 of the required 43 M-Ps have already signed on.
Mr Turnbull is also seeking advice from the solicitor-general on the eligibility of Mr Dutton to sit in the parliament.
Mr Dutton is being questioned over his links to child care centres which received public money as of July 2018.
But Liberal Senator Zed Seselja has told Sky News, Mr Dutton is confident he is eligible.