Mr Kerry and his Russian counterpart have spoken by phone
The US has warned Russia that it will not set up a promised military cooperation project until Syria's regime forces allow aid into besieged cities.
America's Secretary of State John Kerry and Russia's foreign minister Sergei Lavrov reached a deal last week that included allowing the delivery of humanitarian aid to starving civilian areas such as Aleppo.
Under the ceasefire, the US and Russia said they would set up a "Joint Implementation Centre" to share targeting information for airstrikes in Syria if the truce that took effect on Monday holds for one week.
This would see the two Cold War foes fighting on the same side for the first time since World War II.
But the ceasefire is shaky, clashes have continued and aid has not been able to reach Aleppo.
The UN's humanitarian affairs spokesman Jens Laerke said: "In order to actually initiate the actual movement of these convoys we need the facilitation letters. They have not come.
"It's highly frustrating."
In a phone call between Mr Kerry and Mr Lavrov on Friday, Mr Kerry criticised the "repeated and unacceptable delays of humanitarian aid".
State Department spokesman John Kirby said: "The secretary made clear that the United States will not establish the Joint Implementation Centre with Russia unless and until the agreed terms for humanitarian access are met."
Mr Kirby said that Mr Kerry had "emphasised that the United States expects Russia to use its influence on the Assad regime to allow UN humanitarian convoys to reach Aleppo and other areas in need".
Russia complained earlier on Friday that only its ally, the Assad regime, was respecting the ceasefire but they also suggested it be extended by a further 72 hours.
They called for Washington to use their influence over the opposition fighters to get them to abide by the ceasefire.
Reporter Dominic Waghorn said that these issues would be at the forefront of the UN General Assembly meeting next week, along with a new UN report about the Syrian regime's use of chemical weapons such as chlorine gas.
He said: "This UN report is the most damning so far because it not only confirms these claims, using western intelligence and other sources, but it specifies the military units the Syrian regime is using to carry out these attacks.
"The Americans need to negotiate with the Russians, who are allies with the Syrian regime and are supplying the parts and backing for these aerial assets to drop these chemical weapons onto civilian populations.
"I think the stage is set for a major showdown next week at the UN General Assembly, where the west will have to come up with a pretty tough response to this UN report."