"The foundation was Teddy's brainchild. Jerusalem is the poorest city in Israel because a huge proportion of the properties are exempt from tax: synagogues, churches, mosques, education buildings.
"The city doesn't receive anything like the funding it should do. He acknowledged there was a shortfall."
Mr Kollek ensured the foundation focused on wealthy individuals, using the city's religious and historic appeal to encourage them to donate substantial sums for explicit ventures.
Big money meant big names, and when Lord Leigh was first invited to join the board 25 years ago, the foundation was led by some of British Jewry's most notable figures. Its chair was Labour peer Lord Bernstein, with support from Lord Rothschild, Lord Moser, Sir Harry Djanogly, Lord Weidenfeld, Dame Vivien Duffield and others.
"It was really the great and good," Lord Leigh said. "They were all extremely generous but didn't want to go out fundraising. They pretty much funded it themselves with a few mates."
The Jerusalem Foundation's approach remains the same - quietly asking a small number of people to fund big projects. On one occasion Lord Leigh secured a $10m donation from one individual to build a retirement home.
But as the charity marks its half century this year, its new UK chairman is conscious of the need to encourage a new generation of donors to dig deep.
"We are going to change gear," the 56-year-old peer explained. "There are a number of people who will come on board who are from the next generation of the original founding families. When I joined I was the youngest person on the board -– and I still am.
"We are going to bring in people of my age group and younger who are looking after their family trust or have had reasonable success and it will be the same philosophy of identifying projects for the benefit of all in Jerusalem."
Senior partner at the Cavendish Corporate Finance firm which he set up in 1988, Lord Leigh is also a former Jewish Care trustee, a Jewish Leadership Council vice-president, Westminster Synagogue president, and Institute for Jewish Policy Research president.
So how does he assess Anglo-Jewry's efforts to encourage younger donors to grow into the roles vacated by the leading lights of previous generations? "I'm very optimistic. It depends what you mean by 'younger' people - the definition of young in Jewish charities is not having grandchildren.
"There's a huge new wave of people who have made a lot of money in hedge funds, property, tech - the challenge is to keep them connected to the community. Every generation has always worried that the next generation won't follow them, but they do get there."
British-based funding continues to flow to the foundation, and it has helped back a series of cultural and educational facilities, including a basketball court in the Old City, Jerusalem's biblical zoo, and its botanical gardens.
Donors are encouraged to give to projects which have a personal link. Lord Leigh supported the creation of a school carpentry workshop; his grandfather, and then mother, had run a family furniture business.
Among the 4,000 projects have been literacy programmes for Ethiopians, health and welfare provision for Arab Israelis, and the Max Rayne Hand-in-Hand bilingual school educating Jews and Arabs side-by-side.
"We do a lot of work in Arab Jerusalem," Lord Leigh said. "Teddy's view was that if you treat citizens equally and invest in them all equally they will not feel aggrieved, they will feel comfortable. That is our philosophy."
Married with two children, Lord Leigh was ennobled by David Cameron in 2013 after spending more than a decade in senior positions in the Conservative Party, mainly as treasurer or senior treasurer, the role he now holds.
He is one of several Jewish Tories who help form Mr Cameron's view of the contribution Jews make to the country. The community has rarely, if ever, had it so good, Lord Leigh believes.
"Cameron has a great empathy with people of faith; he has his own faith. He is a natural Zionist and he is not alone in the cabinet. When the JLC went to see the Prime Minister there wasn't, frankly, anything they wanted that they didn't get. The community is in safe hands while Cameron, or anyone in the cabinet who I know, is around."
Lord Leigh will next month help lead a cross-party delegation of peers to Israel for the first trip of its kind "in living memory", he said. "They will be unaligned peers, so not known for their Zionist or anti-Zionist interest. The challenge is to get people with moderate views to speak up [in the Lords]. There has been a lot of anti-Israel comment in the House."
As the foundation prepares for its 50th anniversary gala this year, Lord Leigh believes it is on the right track. "Every Jewish person wants to have a connection with Jerusalem. It pulls at the heartstrings," he said. Tellingly the peer concludes that the job of the city is to be "a quiet and constant presence" - much like the foundation that bears its name.
