The trade body said Alison White, who is overseeing the register, made the request at a meeting with the CIPR Public Affairs Group on Wednesday ahead of a launch of a consultation into the register within the next week.
Writing on the CIPR website, the group’s chief executive Alastair McCapra said: "As Alison pointed out, the legislation requires consultant lobbyists to record details of their clients, but not of the ministers or civil servants they have contacted. This is a strange flaw, and because of it, the register cannot provide the level of transparency that was promised when the new legislation was debated in Parliament.
"She believes that for the law to have the intended effect, lobbyists should disclose whom they have contacted as well as whom they are representing."
McCapra added: "She was at pains to point out that she has no power to require this, and she will rely entirely on the profession to deliver this voluntarily. So the ball is now in our court. The formal consultation will launch within the week, and by the end of the year she will want an answer as to whether or not we are willing to assist her by volunteering this information."
McCapra said the answer to White’s request "needs to be a resounding ‘yes’".
"The case for disclosing the information she is requesting is clear and the public interest is undeniable. We should act in the spirit of the legislation and aim to exceed what is expected of us, which will do more to create a culture of transparency and openness around lobbying, with all the benefits to the industry’s reputation that will bring."
He highlighted concerns about "another regulatory burden" for the industry. "It might be that clients are potentially driven away from the searchlight of the register to look for consultants who will help them find ways of avoiding the public eye; and consultancies will have quite a lot to do to meet all the new statutory requirements without adding anything else."
But McCapra added: "The CIPR needs to think long and hard before beginning our relationship with the new registrar by saying ‘no’.
"Public confidence in lobbyists is low, and how the institute responds will help determine whether that confidence is further undermined or in any way improved. It’s also pretty clear that there will be a need, before long, to rectify the various anomalies of the recent Act with a new law, and we can be certain that what was overlooked in the first instance will not be forgotten a second time.
"We can go willingly now or be pushed later, and we have to decide whether we really want to the public to take our protestations of commitment to transparency and the public interest seriously or not."
Plans for the registrar have come under criticism from the PR industry. There was widespread dismay in the sector that the register would not include in-house lobbyists, for example.
The House of Commons Political and Constitutional Reform Committee also expressed concerns about the appointment of White, the former commercial director of Royal Mail.
A report from the committee in September "raised the candidate’s lack of familiarity with the lobbying industry and lack of a working knowledge of Parliament" but added that it was "satisfied" that White had the "professional competence and personal independence" required for the role of lobbying registrar.