Barrow and Furness MP John Woodcock said: "I welcome this promise of swift action from Cumbria police which could bring home the appalling human cost of the cover-up at the CQC. Lives may have been lost... because of people covering their backs rather than acting to protect patients."
Cumbria Police says it will spend three weeks examining the CQC's report into the alleged cover-up into the deaths of mothers and babies at Furness General Hospital.
A statement by the force said:
The former deputy chief executive of the Care Quality Commission, Jill Finney, has insisted she and colleagues did not order the destruction of a report criticising the NHS regulator's investigation at Morecambe Bay NHS trust.
A former director of operations at the Care Quality Commission claims he was sacked after raising serious concerns about the way it was run.
David Johnstone told the BBC he was escorted off the premises, then hit with a gagging order, after trying to introduce fundamental changes in the organisation.
Speaking on BBC Radio 4, the author of the Mid-Staffs report - Robert Francis QC - said there appeared to be "considerable echoes" in the Morecambe case of the findings of his report into the Mid-Staffs scandal.
He said: "There has been apparently a lack of candour and openness attrust level, there was a failure of a regulator to find out that things weregoing wrong and it appears that there could well have been, in many differentplaces, a lack of openness and transparency."
Practices such as redacting and suppressing critical reports "can become self-justifying", he said.
"There becomes a need to keep things quiet to try and sort them out. Whereas actually in something as important as public health we need to be quite open," Mr Francis said.
He added that he had heard and seen evidence of "concerns being raised by individuals at both ends of the organisation being considered very unwelcome by the leadership" of the CQC during his inquiry.
A former deputy chief executive of the Care Quality Commission has denied any involvement in any decision to delete a critical internal report.
Last week, an independent report of the regulator's investigation of mother and baby deaths at a maternity unit in Cumbria found evidence of a "cover-up".
Jill Finney told the BBC: "It was quite clear on reading the report that it was not satisfactory and CQC should have done more. So at that meeting we agreed that the report required much further work.
"There was not a decision at that meeting to delete that report, nor was there an instruction."
She also said that Grant Thornton, the accountancy firm who carried out the review, failed to give either herself or two other colleagues, the opportunity to put forward their side of the story.