Last night the Qatar 2022 bid committee said its members "vehemently deny all allegations of wrongdoing", that, through bribery, the oil-rich state had "bought the World Cup".
But Mr Murphy, who has already highlighted deep concerns over the conditions of foreign workers building the stadiums, said: "If any one of these millions of leaked emails proves corruption in winning the bid, then construction should stop on the stadia, the bid should be switched and the entire contest re-run."
Hundreds of migrant workers have died building the new football grounds.
The MP for East Renfrewshire said, if there were a re-run, at the very least the contest should be open to those countries, which lost out in the 2022 bid, but also to others like South Africa, which successfully held the contest four years ago.
He noted while England might be in a position to bid for any re-run, it was unlikely to win as Russia, regarded as European, was hosting the 2018 tournament.
The Labour frontbencher, a football enthusiast, also said Helen Grant, the Coalition's Sports Minister, should give a Commons statement on developments when Westminster returns from recess this week.
John Whittingdale, the Conservative Chairman of the Commons Sport Committee, insisted there was now a need for an "urgent and full transparent investigation to establish the facts".
He described the position of Sepp Blatter, the president of Fifa, the football's world governing body, as "almost untenable" as he had been very dismissive of the allegations over the past couple of years and did not appear to have taken them seriously.
"If the choice of Qatar was as a result of improper payments being made, then that strengthens an already strong case for re-running the whole 2022 contest," he said.
Qatar insists Mohamed bin Hammam, a former Fifa vice-president, had nothing to do with its campaign to take the 2022 World Cup to Doha and that it disowned him once he was banned from football in 2011 after being caught bribing voters in his campaign to be elected Fifa president.
But pressure on Mr Blatter to get to the bottom of the latest allegations has increased after Jim Boyce, a Fifa vice-president, said he too would be in favour of re-running the 2022 vote if the claims of bribery were proven.
He stressed Michael Garcia, Fifa's chief investigator, who is already probing allegations of corruption, would now have to widen his investigation.
As football fans across the globe prepare for the kick-off next week to this summer's World Cup in Brazil, it emerged a large cache of leaked documents is alleged to show Qatar's victory in securing the tournament in eight years' time was sealed by a corrupt, covert campaign undertaken by Mr Bin Hammam.
It is claimed the ex-Qatari Fifa official used slush funds to make payments totalling more than £3 million to football officials to create a groundswell of support for the oil-rich Gulf state's bid.
Mr Bin Hammam was said to have used as many as 10 slush funds controlled by his private company to make dozens of payments of up to £120,000 into accounts controlled by the presidents of African football associations.
They held sway over how the continent's four executive members would vote.
In December 2010, it was announced Qatar had beaten off the challenges from America, Australia, South Korea and Japan. Concerns were immediately raised about staging the event in a country with no footballing heritage and in temperatures topping 50ºC.