"We have partnered with a local NGO to address the immediate needs of victims, including the provision of emergency food aid to families. This initiative began in Bangladesh immediately the extent of the disaster became clear," the firm, owned by Associated British Foods Plc, said in a statement.
"Primark will also pay compensation to the victims of this disaster who worked for its supplier. This will include the provision of long-term aid for children who have lost parents, financial aid for those injured and payments to the families of the deceased," the statement added.
Among the factories was New Wave Style, which supplied Primark, while some of the other international brands included Matalan, Mango and Benetton.
"Primark notes the fact that its supplier shared the building with those of other retailers. We are fully aware of our responsibility. We urge these other retailers to come forward and offer assistance," a Primark spokesperson added.
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Canadian counterpart Loblaw Companies Ltd, which had some of its Joe Fresh clothing line manufactured at the building, said it too was offering compensation.
Italian multinational Benetton admitted this week that it had been supplied for a one-off order by one of the factories in the building, after previously denying any links with the building.
Bonmarche, a retailer with more than 200 stores across the UK, has acknowledged that its products were made in the block, while the UK chain Matalan also said it had been supplied from the factory in the past.
A petition drawn up by the National Garment Workers' Federation, which called on Primark, Matalan and Mango to pay compensation to victims' families and to sign up to the Bangladesh fire and building safety agreement to prevent future deaths of garment workers, has so far received nearly 600,000 signatures.
Protesters also gathered outside some stores in London over the weekend to call on the retailers to take action.