Robert Berry, 42, from Berkshire, collapsed at the finish of the event on Sunday.
His page on the fundraising website JustGiving shows that donations to the charity have passed £40,000 – more than 20 times his original target of £1,700. More than 3,500 donations have been made.
The page says that Berry was running because of the support the charity gave to his mother, who suffers from osteoporosis and has had two sets of bones in her lower back collapse.
Claire Severgnini, chief executive of the NOS, said: "Our immediate concerns are for Mr Berry’s family. Our thoughts and deepest sympathies are with them at this tragic time."
The charity said that it would not release any further statements at the request of Berry’s family.
A statement on the London Marathon website says: "The organisers of the Virgin Money London Marathon would like to continue to express their sincere condolences to the family and friends of Mr Berry and our thoughts and deepest sympathies are with them all at this difficult time."
Donations to the emotional support charity Samaritans in memory of Claire Squires, who died while running the marathon for the charity in 2012, reached almost £1m.
JustGiving said it would donate any fees on the processing of donations made through Berry’s fundraising page to the charity, as it had done in the case of Squires and others.