It said: "Minister Shoygu 'guaranteed' that there were no Russian military personnel involved in the humanitarian convoy, nor was the convoy to be used as a pretext to further intervene in Ukraine."
It said: "Minister Shoygu assured Secretary Hagel that Russia was meeting Ukraine's conditions."
Ukraine insists on a full inspection.
Border guards have arrived at the convoy but have not yet cleared it to pass.
The BBC which has a journalist following the convoy, says some of the lorries were opened up for the media on Friday, and the main thing that struck him was how empty many of the vehicles were.
The Red Cross Head of Operations for Europe and Central Asia. Laurent Corbaz, said: "The agreement... foresees that the trucks will be checked by the Customs officers of Ukraine on Russian territory and then be allowed inside Ukraine with ICRC people accompanying the convoy and then unload somewhere in Luhansk and go back to Russia, empty."
He said that given the size of the cargo and the security constraints, "implementation is likely to take some time", not in one week and possibly "much longer".
Russia has complained about Ukrainian operations in the area.
The Russian foreign ministry said: "We draw attention to the sharp intensification of military action by Ukrainian forces with the apparent aim to stop the path, agreed on with Kiev, of a humanitarian convoy across the Russia-Ukraine border."
