The Syrian diplomat was not named, but Damascus denied the transfer, saying Israel might be using the accusation as a pretext for a military strike against it.
Hostility to Israel
Isreael and Hezbollah, a powerful political-military organisation, fought a major war in the summer of 2006 over the capture of Israeli soldiers.
Their mutual hostility remains strong even after Israel pulled its last troops out of Lebanon 10 years ago.
The US statement stopped short of confirming the alleged transfer of the long-range missiles, which, if true, could undermine the diplomatic overtures to Syria by Barack Obama, the US president.
The state department said this was the fourth time in recent months that Washington has raised the issue with the Syrian embassy.
Hezbollah is backed by Syria and Iran, which started funding the group's campaign to drive Israeli soldiers out of Lebanon in the early 1980s.
The group has replenished its arsenal beyond levels it had in the 2006 war, according to a Pentagon report on Iran's military sent to congress and made public on Monday.
Scud missiles in Hezbollah hands could strike deep inside Israel, while a partial transfer could involve weapons parts, documents or funding, according to US officials.
The officials said last week they believed Syria intended to transfer the weaponry, but had doubts about whether the missiles were delivered fully assembled or had actually been transferred to Lebanon.
The US says Syria's designation as a "state sponsor of terrorism" was tied to its support for Hezbollah.