Cllr Adam McVey said it was a great opportunity for the Scottish Football Association to embrace a new era for the sport in a possible move that would pump millions into the city.
The Scottish FA’s existing Hampden lease is due to expire.
A decision on the future home of Scottish football is due to be taken on Wednesday.
Scottish Rugby has claimed it could pour extra income of around £2 million a year into the Scottish FA’s coffers if Murrayfield succeeded in ousting Hampden as the country’s primary venue for international football.
Cllr McVey said: “We’re backing the BT Murrayfield bid because Scottish Rugby is thriving and has put together a comprehensive and compelling bid to maximise the opportunities that the stadium and the capital city provides.
“As the largest stadium in Scotland, BT Murrayfield routinely hosts some of the country’s highest profile sporting and cultural events, from the Six Nations to sell-out concerts. The extra capacity in and around the ground could generate significant sums for the betterment of the Scottish game, on and off the field, creating a home for our national sports to win together.
“The stadium is just a 20-minute journey from the airport or five minutes to Princes Street by tram, which connects ticket-holders to Edinburgh Waverley and Haymarket train stations and the main bus station in St Andrew Square. The stadium has never been easier to get to.
“Here in Edinburgh we are well used to providing a magnificent backdrop for international festivals and events, attracting millions of visitors each and every year.
“Together with our partners, we boast years of shared expertise and experience in providing the warmest and safest welcome to our beautiful capital city – working hard to ensure visitors’ needs are balanced with those of local residents. This would be developed further to aid the safe arrival of football fans for major matches just as we have in the past for Celtic’s European cup ties, Barcelona friendlies or Hearts’ league matches that saw some 14,000 Rangers fans travel to add their voice to a record crowd.
“We are always looking at ways in which we can improve the overall match-day experience in the city and adding events of the scale of Scotland football internationals and major cup finals to our calendar would undoubtedly add another string to the city’s bow, particularly when you consider the millions of pounds they would generate for the city and wider Scottish economy.
“What a wonderful opportunity for the Scottish FA to embrace a new era for the sport. I hope they make the right decision for the Tartan Army and for football fans across the whole of Scotland.”