Mr Shapps told the Prime Minister in his resignation letter that the “buck should stop with me” over issues while he was Conservative Party chairman.
Grant Shapps is set to resign amid allegations he failed to deal with allegations of bullying within the Conservative Party.
The parents of an activist who killed himself had called on the former party chairman to quit the Government, insisting their son Elliott Johnson would still be alive if Mr Shapps had behaved “responsibly”.
David Cameron said a statement about the international development minister would be made as he told reporters that Mr Johnson’s death was a “tragic loss”.
The party has said it was not aware of any complaints against Mark Clarke, who ran the party’s “Road Trip” operation during the general election campaign, until August.
He has since been expelled over bullying and blackmail allegations, but denies any wrongdoing.
The Prime Minister, who is in Malta for a Commonwealth summit, said: “I feel deeply for his parents. It is an appalling loss to suffer.
“In terms of what the Conservative Party should do, there needs to be and there is a proper inquiry to ask all the questions and interview all the people who have come forward.
“That will take place.”
He added: “It is a tragic loss of a talented young life. It is not something that any parent should have to go through and I feel for them deeply.”
The Prime Minister said an independent lawyer would oversee the inquiry process.