Alex Salmond said he had a positive discussion with the sporting hero and his mother, top tennis coach Judy, about the idea to nurture young Scottish talent.
The Scottish Government will explore the suggestion with the Murrays over the next couple of months, Mr Salmond said.
Murray, 25, who also took the men's singles title in the London 2010 Olympics and with it a gold medal, was given a hero's welcome by thousands of people in a triumphant return to his hometown of Dunblane, near Stirling.
He celebrated a golden summer of success with locals just days into his return to the UK after beating Novak Djokovic to win the US Open.
That victory came just weeks after he beat Roger Federer to take the Olympic men's title at London 2012.
Mr Salmond said he was delighted to meet Murray.He said: "Scotland is very, lucky to have Andy Murray whose legendary status is now secured, and his triumphs at both the Olympics and at the US Open will act as a spur for future generations of Scots.
"Both Andy and Judy are passionate about developing tennis in Scotland to ensure that youngsters have a greater access to the facilities and coaching that they need to make the most of their talent and potential.
"We had a positive discussion about their idea for a tennis academy which is certainly in line with the Scottish Government's ambition to improve young Scots' participation in sport.
"We'll be exploring this with the Murrays over the next two months and we hope to make an announcement in the near future."
The First Minister presented Murray with his official Royal Mail stamp for his medal success at the London Olympics.