Ian Grabiner, who has been Green's chief executive since 2009, is one of about 20 executives of high street clothing companies with supplies coming from Bangladesh who are being invited to a meeting with Greening, where she will urge them to work with the government to raise standards.
In April, 1,127 workers were killed in the collapse of the clothing factory in the Dhaka suburb of Savar. In a second incident three weeks ago, eight people were killed in a fire at a factory in the capital that was producing clothes for western retailers, including Primark.
At least 1,800 Bangladesh garment industry workers have been killed in fires or building collapses since 2005.
Arcadia, which owns Topshop, BHS, Evans and Dorothy Perkins, is yet to sign up an Ethical Trading Initiative (ETI) agreement between companies to impose a code of labour practice on all their suppliers, although Arcadia has agreed to sign an accord on fire and building safety in Bangladesh.
Eight other high-street retailers – Matalan, Aurora Fashions, Peacocks, Shop Direct, Hobbs, French Connection, Karen Millen and Austin Reed – whose executives have been invited to the meeting with Greening, have also failed to sign the ETI and will come under pressure to do so. Arcadia has previously said that it enforces its own code of conduct on suppliers, and that Bangladesh represents only a small percentage of its supply chain.
Greening told the Observer: "The Department for International Development will work with companies to see how we can take the next step, whether it's signing up to ethical trading standards or going the extra mile in ensuring responsible business practices."
The call comes before a visit this week by Alan Duncan, the minister for international development, to Bangladesh, which will include talks with the Dhaka government on how the UK can support improvements in building standards.
An official investigation in Bangladesh found that causes of the disaster included the swampy ground that the factory was built on; the "extremely poor quality" construction materials; and the massive, vibrating equipment operating when the eight-storey building collapsed.
Since David Cameron's cabinet reshuffle, when Greening was moved from transport to international development, she has made economic development a priority for her department. In a speech in March she said: "I want to see an end to aid dependency through jobs. I believe Britain can be a force for good. I also believe British business can be."
A spokesman said: "The government is talking to the Bangladesh government about helping to improve building regulations, but there is more we can do. We want to work with businesses to keep up the momentum for change."
Other companies whose chief executives have been asked to attend the meeting with the development secretary include Sainsbury's, Marks & Spencer, John Lewis, Next, Debenhams, New Look, Mothercare, Monsoon, Accessorize, Asos, Burberry and River Island. A statement on Arcadia's website says: "We do not own or operate factories and Arcadia Group is rarely dominant in an individual factory. We have strong working relationships with our network of international suppliers, 48% of which have been with us for three years or more.
"Arcadia brands work with around 700 suppliers that manufacture our goods in approximately 1,060 factories. The top 20 suppliers provide 44% of our goods."