The newly crowned Wimbledon champion yesterday said he would canvass opinion and could declare his stance ahead of the vote in September next year.
The 26-year-old Scot was speaking in the wake of his greatest triumph when he defeated Novak Djokovic, the world No.1, in straight sets on Centre Court.
"When the time is right I will probably say something about it," said Murray yesterday.
He added: "I'm going to get asked about it all the time. I will think about it, speak to some people and try to see what is best for the country."
Murray, if he declares his stance, would become the latest famous Scot to come out for or against on the issue.
Manchester United manager Sir Alex Ferguson has already donated to the Better Together anti-independence campaign.
Hollywood actor Alan Cumming has always been a high-profile supporter of the Yes Scotland organisation, along with the actor Martin Compston.
Sean Connery is undoubtedly the SNP's best-known celebrity supporter, while Hollywood actor Brian Cox is another backer of independence.
Ewan McGregor, Billy Connolly, Primal Scream's Bobby Gillespie and singer Sharleen Spiteri, of the band Texas, support the No campaign.
However, Olympic cyclist Sir Chris Hoy faced online abuse earlier this year after saying that independence would weaken the British team and, with a lack of adequate facilities north of the Border, would create a challenge for Scotland's new national athletes.
Murray was also questioned on what his response would be to any offer of a knighthood to celebrate winning Olympic gold, the US Open and breaking a run of British failure in the men's singles that stretches back 77 years to the victory of Fred Perry.
"I don't really know. It is a nice thing to have or be offered. Just because everybody's waited such a long time for this, that's probably why it'll be suggested. I don't know if it merits that," he said.
Murray's return to Dunblane, his home town, is almost certain to be delayed beyond the summer.
"I do want to go back," said Murray, who was greeted by 15,000 people when he returned to the town in September after his victories at London 2012 and the US Open.
He said: "I'm not sure exactly when that will happen. I haven't had that much time to think or co-ordinate things.
"I want to go away on holiday and try to get some rest, because the last few weeks have been tough, pretty stressful for me.
"I will try and get a bit of time on holiday then I need to get back into training for the US stretch.
"But I would like to go back to Dunblane."
He also insisted that his growing celebrity would not sidetrack him from tennis.
"I don't really have any distractions," he said. "I'm not like addicted to going out or drinking or smoking. I don't do any of that sort of stuff. I enjoy being around my friends, I enjoy training. I enjoy being over in Miami.
"You see it a lot in other sports because with fame there comes a lot of distractions. But it comes down to the people you surround yourself with. If you surround yourself with the right people you won't get yourself in those situations."
The world No.2 also revealed part of the pep talk given to him by Sir Alex during the tournament.
Murray said: "One of the things he said to me was just that one of the things he built his teams on was consistency and concentration. If you can concentrate throughout an entire match you'll gain that consistency.
"That was something I tried to do throughout the tournament. It is something I try to do but it kind of emphasises the point a bit more when someone like him says it to you."