The Met said prosecutors had advised them that the "lack of systems" in Parliament would "severely undermine the viability of any criminal investigation".

The Old Bexley and Sidcup MP was severely reprimanded by the Commons authorities in January following allegations that he paid two sons tens of thousands of pounds from the public purse despite them having done little apparent work.

Mr Conway has since been stripped of the Tory party Whip, and announced he will be standing down at the next general election - although he will still draw his £60,000 salary until then.

The scandal sparked renewed outrage over the level of public funds available to Parliamentarians in expenses, and the lack of transparency and accountability.

Last month Met Commissioner Sir Ian Blair suggested that he backed a police probe into Mr Conway's actions, insisting that police would not "ignore the situation".

However, in a statement the Yard pointed out that Commons sleaze watchdogs had already looked into the situation, and MPs had voted to suspend Mr Conway for 10 days.

"The allegations have already been investigated by the Parliamentary Commissioner and then put before the Parliamentary Committee on Standards and Privileges who recommended sanctions that were imposed by the House," the statement said.

"The CPS advised us that they are of the view that the lack of systems in this case to account for MPs expenses would severely undermine the viability of any criminal investigation leading to a prosecution. In these circumstances we do not believe that it is appropriate for a police investigation to be instigated."

The office of Commons Speaker Michael Martin said they would not be commenting because the case was a police matter, but stressed that an urgent review was under way into the level of MPs' expenses and the systems for scrutinising them.