Lord Feldman is under fresh pressure to quit as Conservative Party chairman over the handling of bullying and blackmail allegations.
The row over alleged behaviour by Mark Clarke shows little signs of abating despite the resignation of Grant Shapps, who was co-chairman with the peer until the general election.
The departure of both men has been demanded by the father of a Conservative Party activist who killed himself, leaving a note condemning Mr Clarke - who denies any wrongdoing.
The final straw for Mr Shapps appears to have been the emergence of an email sent to him by ex-minister Baroness Warsi in January, complaining that the election campaign aide had been abusing her on Twitter.
The party had previously insisted no complaints against Mr Clarke, who ran its RoadTrip canvassing drive in the run-up to the election, were received before August. He has since been expelled.
In a letter to David Cameron confirming his resignation as international development minister, Mr Shapps said he had concluded the "buck should stop with me".
He wrote: "Although neither the party nor I can find any record of written allegations of bullying, sexual abuse or blackmail made to the chairman's office prior to the election, I cannot help but feel that the steady stream of those who raised smaller, more nuanced, objections should have perhaps set alarm bells ringing sooner."
Downing Street has stressed that Lord Feldman retains the "full confidence" of the Prime Minister.
But a senior Tory source confirmed the peer is among 40 "witnesses" who are giving written evidence to party officials.
Lord Feldman is understood to have launched the investigation into the allegations surrounding Mr Clarke in August, but has never played a part in it.
Clifford Chance LLP has been instructed to prepare a report on the issues raised, with a remit to assess whether complaints were handled properly and "identify any individuals who were at fault".
The firm's lawyers will also consider the integrity of the evidence-collecting process and whether the right people have been interviewed.
The collection of evidence is not expected to be completed until the end of the year, and Clifford Chance will produce the report "as soon as possible after that".
The source added: "It will be a decision of the board which parts of the report to publish having regard to protecting vulnerable witnesses, the ongoing coroner's inquiry and police investigation. Subject to this, the party intends to publish the key findings and recommendations."
Defence Secretary Michael Fallon said Mr Shapps had taken responsibility for the scandal but stressed the party needs to see "where the investigation takes us".
He told BBC One's Andrew Marr Show: "The person directly responsible for central office, for campaigning, the co-chairman Grant Shapps, who signed up Mark Clarke's operation - he has accepted responsibility and yesterday he resigned."
Asked whether Lord Feldman should resign, he said: "It was Grant Shapps who was in charge of campaigning and who was involved in this and I think it's best now to see where the investigation takes us."
Senior Tory backbencher Bernard Jenkin called on Mr Cameron to launch a "thoroughgoing" governance review of the party.
He said the party's board was too big and unwieldy, and there was a need to examine "the structures and the values by which we operate, the mission for the people who work and volunteer in the Conservative party and how we lead them, and what we expect acceptable behaviour and attitudes should be".