Mr Murray, who resigned as the Shadow Scottish Secretary after criticising Labour leader Jeremy Corbyn, won the seat by around 2,600 votes in 2015.
His shock win, even as the SNP swept the board almost everywhere in Scotland, was thought to be down to a high personal vote and the scandal that affected the SNP candidate, who was revealed to be a ‘cybernat’ who had used a Twitter account to insult pensioners.
Mr Murray, according to the Guardian, said he supported tactical voting if voters’ key concern was defeating the SNP.
He told the paper that Tory and Lib Dem supporters keen to oppose the SNP needed to back him, adding: “If people are saying they want to protect the union, the candidate in the best position is me.”
Mr Murray was sharply criticised, with many pro-SNP commenters online feeling he had implicitly endorsed voting Conservative in areas where Labour could not defeat the SNP.
Scott Rogers, a Labour activist and council candidate, wrote: “It’s dreadful that our sole remaining MP will use his platform to sell out on people suffering under the Tories to protect a seat.”
The MP hit back at that suggestion on Twitter, saying he had suggested that people vote Labour in his constituency and the country.
Labour are confident of holding the seat, as they also won the Edinburgh Southern seat at the Holyrood elections last year.
The SNP haven’t selected a candidate as of yet, but Doug Thomson, a prominent local activist and the husband of political journalist Mandy Rhodes, has put his name forward.
The Party’s Westminster chief Angus Robertson MP said: “This is an extraordinary plea for a Tory election pact from Labour’s last remaining MP in Scotland, whose priority in this election is saving his own job at the expense of the whole country.
“Ian Murray is utterly apathetic to the prospect of keeping the Tories out of Downing Street – having made it clear that he doesn’t see Jeremy Corbyn as a legitimate candidate for Prime Minister while appealing for people to elect more Tory MPs in an effort to keep the SNP out.
A Labour spokesman defended Murray, saying: “In constituencies across Scotland, it is only Labour that can stop the SNP.
“If voters want to stop the SNP and its plans for a divisive second independence referendum, their only option is to vote for Labour.
“Up and down the country, even those who may have not voted Labour before will know that we are the only party that can stop the SNP.
“In Edinburgh South, it is a straight choice between Ian Murray, who will fight tirelessly against independence, or an SNP candidate who will only be interested in breaking-up Britain.
“A vote for the Tories would be wasted, and would risk letting the SNP in. Edinburgh South doesn’t need another backbench Tory MP cheerleading the campaign for a hard Brexit or another SNP MP cheerleading the campaign for Scottish independence.”