Aid agencies are trying to beef up relief operations across Syria, where the ICRC says that needs have grown "exponentially" in the past few weeks due to the escalation of violence in the 17-month-old rebellion against Assad.
Clashes and continuous bombardment have cut off many civilians from basic services and life-saving supplies.
On Tuesday, Syrian President Bashar al-Assad told Peter Maurer, the head of the ICRC, that he welcomes humanitarian operations carried out by the organization in Syria – as long as it works “independently and neutrally,” reports Syrian newspaper Day Press News.
Syrian official Ali Haidar said caution was called for in allowing in aid workers, as such organizations could be making an "impermissible request to open doors that violate Syrian sovereignty," reports Reuters. A diplomatic source also said that "Syria has been very unwilling to grant access and independence to the ICRC once they get in."
For those displaced by the conflict, finding safety, food, and water is increasingly precarious, according to a report by the BBC.
Every family has a story to tell – stories of fear and horror, of blood and loss.
One of the men, Abu Salem, says he fled the central town of Rastan with his wife and four children four months ago after a rocket hit their house.
They travelled to Damascus and ended up renting a 50 sq m (538 sq ft) flat in one of the capital's suburbs. Then early last month, the area came under heavy bombardment from government forces.
Abu Salem says they yet again had to flee, but this time there was no place to go except a nearby public park. They spent 12 days there with many other families until some aid workers found them a place at the old house.
"The neighbours provided us with some food, but we spent 12 days without a shower," he recalls.
Abu Salem used to work in construction but – like hundreds of thousands of other Syrian men – he has not earned anything since protests against President Bashar al-Assad began in March 2011. Like hundreds of thousands of other Syrians, he and his family have also been displaced from their homes by the ensuing conflict.