It's what we are good at: art and culture is something we do well

Sitting in shirt and tie in his peaceful office in the old Dean Gallery, now known as Scottish National Gallery of Modern Art Two, the tall, understated, Belfast-born director says that, at first, he suspected the ominous letter from Her Majesty was a final tax demand. The comment is a sample of his quiet but cutting humour. In fact, the knighthood honoured his seven years' service for the NGS, as well as 10 years working at the National Gallery in London, and his stint as general director of the Van Gogh Museum in Amsterdam, where he was the first foreign-born director. The knighthood is not his only gong. In 2005, Leighton was appointed Chevalier in the French Ordre des Arts et des Lettres, he is a Fellow of the Royal Society and has an honorary doctorate from Edinburgh University for his "outstanding contribution" to art and culture.

But Sir John says of his prospective date with the Queen: "It came out of the blue. It's not something you look for, or work for, or is expected. You do think about accepting it – and then you think the wrath of your 90-year-old mother would be impossible to face if you turned it down."

Sir John, born in 1959, has had a busy, stressful, but ultimately satisfying time in Edinburgh, where he attended university after first being educated in his native Northern Ireland. During his tenure, the NGS has managed to achieve record visitor figures, unveiled the upgraded Scottish National Portrait Gallery and secured the significant contemporary art collections of Anthony d'Offay as the nationwide touring Artist Rooms scheme. That remarkable collection of contemporary art has now been seen by some 16.5 million visitors in 92 exhibitions across the UK. And, perhaps the source of the most stress in his tenure, he made sure the Duke of Sutherland's Bridgewater Loan, the single most important private collection of Old Master paintings on loan to any institution in the UK, was not lost to Scotland.

Two mesmeric works by Titian – Diana and Actaeon and Diana and Callisto – are now safely part of the national collection after Leighton led moves to raise the £100m required to secure them. He nods at the list of achievements but adds: "There is still so much to do. I had a feeling last year a number of projects were coming to an end, so we are now spending a lot of time looking forward."

Sir John has at least one more major project to complete: the re-housing and re-hanging of the Scottish collection of pictures at the National Gallery. When he joined the NGS in 2006, he said the display of Scottish art in its windowless 1970s' concrete basement resembled an "apology" and he was "deeply unhappy" about it. Now he is planning to change the gallery on The Mound and its displays. The specific ways of doing this are evolving as the architectural footprint of the gallery itself –hemmed in by Princes Street Gardens, the railway line and a road – is limited.

"One of the enduring weaknesses in recent times has been the display of the historic Scottish collection," he admits. "When I came here, it was obvious the National Portrait Gallery, in terms of hardware, was the top priority. But we are now looking very closely at the Scottish National Gallery, and that will be a very different project. There will be phases of improvements. The roof is phase one, and phase two is about looking at that whole basement space. The fact you call it a 'basement' sums it up."

He adds: "What you want is a space where you can show the best of Scottish art to a local, national and international audience with pride, and where you can also set up interesting dialogues with wider western European art traditions. And we don't have that at the moment.

"Yes, it will cost money but one of the things we've seen is that investing in these assets generates great returns. In the wider economic world we find ourselves in, it is tempting to just shrink and to do less, to stop doing certain things, and certainly we have found that, if you can get the capital investment, and regenerate and re-invent, the returns are great. We won't be roofing over part of Princes Street Gardens, but there will be something that adds a huge amount of impact for our visitors."

He has a small collection of newspaper cuttings in a neat file, marking the stages of his career. He is still tickled by an early write-up which said he was the top man for a job "even though he was born in Belfast". Also there, on yellowing paper, he is pictured with a full head of hair on the steps of the National Gallery in London, inside the Van Gogh Museum, and then in The Herald where he is described as the "Van Gogh Man". He laughs at this and thinks back to his first days on the job: "If we imagine ourselves sitting back then, it was obvious the NGS has an amazing collection, and it really is. You become ever more aware of how blessed the country is with the collection. It has great staff, great people and an eager public, a fantastic setting, so [my job] was about building on strengths. But what pretty quickly became clear was the sense the NGS was too Edinburgh-centric. And I think we have gone some way to addressing that. Artist Rooms has been key to that. I think the relationship with the public could still become more porous. There is room to build a wider, deeper, affection for both the collections and what they are about, and build that sense of ownership."

He has appointed a new director of public engagement, "an Orwellian title" he laughs, to help make this fine art more accessible, and is keen to see galleries accepted as rare, "safe social spaces" and mentions the "affection" the public in Glasgow holds for the Kelvingrove – and would like to see that in Edinburgh too.

However, his most difficult time in the past seven years was the prospect of losing key paintings from the Bridgewater Collection, the Duke of Sutherland's long-term loan that forms the core of the National Gallery's Old Masters.

When the threat first loomed in 2007, he and his chairman described the then-secret threat as Project Brandy –"because that's what our chairman Ben Thomson and I needed when we first heard the figures involved. It was 'okay: we need £100m'." There were times, he admits, when he thought the fundraising battle might have been lost, especially following the financial crash of 2008. Now the paintings are secured for the long term in a sharing deal with the National Galleries in London.

He says: "Before I came here I was aware the profile of the Scottish National Gallery was so dependent on those pictures. But it was never just about those paintings, incredibly beautiful that they are. It was about the collection and it was also about Scotland as a world-class destination. How would you explain it to future generations if you lost it? Overall I felt that, if you take the long view, in 100 years' time, people would look back, the sums of money would recede and you are left with these absolutely fantastic paintings for people to enjoy."

Sir John, whose son is at Stirling University while his daughter is studying in Amsterdam, has no intention of moving. He tells me to return in seven years and see the fruits of his further labours. One of these will be properly displaying and explaining the modern Scottish artists, largely from or based in Glasgow, who have been so successful in the contemporary art world. "It's something we are good at: art and culture is something we do well," he says.

"Art is part of the solution, not part of the problem. You have this tremendous success for an array of artists, yet their work is not that well known here. There's a job of work to do, because it is not as if the art is inaccessible. It's very human, very lively, witty and engaging – there's no reason why you can't have a wider public for it.

sir john leighton Director, National Galleries of Scotland