The world number two, who has Wimbledon, US Open, Davis Cup and Olympic singles titles to his name, is still focused on claiming so-far-elusive crowns at the Australian Open and Roland Garros.
But it is the French Open which Murray believes would represent his biggest achievement, given the unique challenges of the red clay.
“Clay has always been the most challenging surface for me since I came on the Tour," Murray told Eurosport. "But I’ve also kept improving on it and my results at the French Open over the last few years have got gradually better.
“If I was able to win the French or even reach the final then it would be a big, big achievement for me because a few years ago I didn’t know if that was going to be possible. I’ve kept trying to improve on the surface and my results have got better the last few years.
Winning the French Open, because it’s the most challenging surface for me, would be my biggest achievement - so I’m very motivated for that event this year and it’s the start of a really important summer.
“Clay has been the surface I’ve had the least success on: it’s just the reality that I haven’t been as good on it. Until the last 18 months or so, I hadn’t beaten many of the top 10 players on clay before - but that changed last year. I’m hoping I can do a little bit better again this year.”
Murray has met Novak Djokovic in six Grand Slam finals with the Serb winning four times in Melbourne Park showpieces, while the British number one claimed memorable wins at the US Open and Wimbledon, and he counts him as his toughest opponent, along with Rafael Nadal.
“Novak [Djokovic] and Rafa [Nadal], in terms of results, have been the toughest. I’ve had some important wins against them in Grand Slams, but I’ve also had a lot of tough losses against them - long, close matches.
Over the last couple of years, Novak has been the clear number one and the toughest opponent I think for most of the players.
Britain's Andy Murray and Serbia's Novak Djokovic pose for a photo before their final match at the Australian Open
Image credit: Reuters
Most unexpected achievement
Murray inspired Great Britain to a first Davis Cup win since 1936 last year and became only the third player to win the maximum eight singles matches - a triumph he says was the most unexpected of his career so far.
“That [Davis Cup triumph] was the most unexpected. None of us in the team thought that we would win the Davis Cup and it had also been a much longer process to get to that winning moment.
“It was five years from when we were way back in the competition, and then all of a sudden we were in the quarters, semi-finals and we started to think anything could happen.
“It was a long process and obviously to do it with my brother made it extra special so I was proud of that one because it’s not easy to do. You have to rely on a lot of people during the Davis Cup, not just on yourself. Everyone in the team stepped up and helped us win.”
Davis Cup format
“I think the one thing that needs to remain in the Davis Cup is the home and away aspect. If you held it in one city, I don’t think the atmosphere or matches would be as good. You wouldn’t get as big crowds coming.
“Last year, we played in front of packed crowds in all of the matches so I think that has to remain. But I do think that it’s good to look at different alternatives that could make it more successful and have more of the top players playing more often.”
Murray won gold for Britain at the London 2012 Games, losing just seven games to Roger Federer in the final, and he says he will never forget the impact of that success ahead of the Rio Olympics this summer.
“For me, winning a home Olympics was huge. That was the biggest competition I had won to date. I’d never won a Grand Slam before that and it was amazing to win a gold medal in London.
It was something I had never expected to get the chance to do – it was a once-in-a-lifetime opportunity. That meant a lot to me.
“It’s different to the Grand Slams. When you’re competing for your country it adds a little something and all the top players are playing at the Olympics in tennis now. It’s a huge event.”
Current sporting inspirations
“I like basketball and Steph Curry has been the best player for the last couple of years over in the States – I love watching him.
“Footballers – I love watching [Lionel] Messi. I think he is an unbelievable talent; and [Cristiano] Ronaldo.
Stephen Curry #30 of the Golden State Warriors dribbles the ball against Zach Randolph #50 of the Memphis Grizzlies in the second half during the game at ORACLE Arena on April 13, 2016
Image credit: AFP
“And tennis as well. I love watching tennis. Any of the top players across the men’s and women’s games I enjoy watching and I try to learn from as well.”
Life after tennis
“I’m not sure what I’ll do when I finish playing. I enjoy being a professional sportsman and I love tennis so I hope I still stay involved in some way, but maybe not for the first few years when I finish playing as it would be nice to try to do something different and challenge myself at something else.
“I think in the long run I’ll still remain in tennis because it’s the sport that I love and a lot of my friends and family are involved in it as well.”
Favourite location on Tour
“I love going to Australia and Melbourne – it’s a great city. I love going to Miami and spend a lot of time there with my training. That’s the thing with what we do: we get to go to pretty much all of the big cities in the world. We do get to see all of the great, big cities. I love Paris and Rome is a really nice place to go to.
“It’s hard to pick one, but I do always love going to Melbourne at the beginning of the year – it’s a bit different over there, great weather, the people are good fun, they love their sport.”
Favourite Grand Slam
“Because they [Australian Open] are the first one of the year, they are always making improvements which the other Slams then follow or try to beat, but they always seem to be the leaders. Every year they make changes to the tournament, to the facilities and it’s a just a real players’ tournament.
“It’s fantastic and just an easy event for everyone – that’s why I love it. Wimbledon for me is very special because it’s my home Slam and it’s 20 minutes away from my house. It’s obviously got so much history too, but Melbourne is a great, great way to start the year, it’s a really good tournament.”
Britain's Andy Murray with brother Jamie at a training session at the Australian Open in Melbourne
Image credit: Reuters
On Tour as a new dad
Murray and wife Kim became parents in February and he says it has been a big - but very happy - adjustment to make with daughter Sophia travelling to ATP Tour events in recent months.
“It’s obviously been great the last few months. I’ve loved it; it’s been a huge change in my life and it takes a little bit of time to adjust to that as well.
“She’s [daughter Sophia] travelled to a couple of the tournaments, to Miami and Monte Carlo, and it’s been great having her around and having my wife around more as well because of that.
“I’ve got to spend a lot of time with my family the last few months which is nice. It’s obviously a big change, a big adjustment, and it takes a bit of time to get used to.
I’ve really enjoyed it so far and everyone’s told me it just keeps getting better and better, so I hope that’s the case!
‘Crowd on their feet, players nearly on their knees’ – Murray wins deciding game
Andy Murray was speaking to Eurosport as part of an Under Armour event.