Such an approval process is in line with diplomatic practice.
Crowley said it is premature to make a formal announcement pending a reply from Damascus and Obama's own move to name publicly his choice for envoy to Damascus -- which also requires approval from the US Congress.
Neither US officials nor Muallem would disclose the nominee's name.
A State Department official, who asked not to be named, told AFP that Middle East envoy George Mitchell submitted Washington's nominee for ambassador to the Syrian authorities when he visited Damascus two weeks ago.
Since his appointment a year ago, Mitchell, a former Northern Ireland peacemaker, has made a number of visits to Syria during regional tours to neighbors Lebanon and Israel as well as the Palestinian territories.
A key regional player, Syria backs anti-Israeli militants in Lebanon and the Palestinian territories but has also engaged in peace talks with the Jewish state in a bid to recover the Golan Heights, captured by Israel in the 1967 war.
It was not clear how much progress Mitchell has made in trying to get Syria to drop support for the militants.
"We have questions for Syria in terms of its support, even current support, of extremist groups in the region," Crowley said.
"But we are committed to advance our relationship and we're committed to work with Syria, Lebanon, Israel, the Palestinians, in pursuit of Middle East peace," he said.
The Obama administration announced in June last year it would send an ambassador back to Syria as Washington tries to engage with Damascus in a bid to revive Arab-Israeli peace talks.
The previous administration of president George W. Bush recalled the US ambassador from Damascus and put relations with Syria on hold in 2005, following the assassination of former Lebanese prime minister Rafiq Hariri.
Hariri's killing was widely blamed on Syria, although Damascus has steadfastly denied any involvement.
Analysts also say US moves to woo Damascus could isolate Syria's non-Arab ally Iran, which is accused of not only backing anti-Israeli militants in Lebanon and the Gaza Strip but also anti-US militants in Iraq.
Speaking in Damascus, Muallem described the role of Washington in the Middle East peace process as "essential, given the strategic relations between the United States and Israel," the close US ally.
"Syria holds out hope for its role, as long as the United States wants its role to be constructive," Syria's top diplomat said.
Muallem also hit back at remarks from Israeli Defense Minister Ehud Barak who said Monday: "In the absence of a peace agreement with Syria, we might find ourselves in a forceful conflict that could lead to an all-out war."
The foreign minister said: "Israelis, do not test the power of Syria since you know the war will move into your cities."