FBI Director Robert Mueller (search) heads to the host city of next year's Olympics (search) on Thursday with a dual agenda: praise Greek authorities for fighting domestic terrorists but press Olympic planners to fill perceived security gaps.

Mueller's two-day visit, scheduled to begin Thursday, highlights the gap between Greek confidence in the Olympic security blueprint and Washington's insistence that more needs to be done.

Mueller is expected to address one key U.S. worry: the relentless flow of illegal immigrants into Greece from Balkan neighbors to the north and by sea from the Turkish coast to the east. There are fears terrorists aiming at Olympic targets could possibly use the clandestine routes.

Other points of friction include how to protect the American team during the Aug. 13-29 games. At least 100 U.S. security personnel are expected to watch over the athletes, coaches and others. Greek officials have said the agents cannot be armed, but it's unclear whether this applies to only Olympic venues or all areas of the city.

Greek officials say the security plan for Athens (search) is comprehensive and being fine-tuned with the help of a seven-nation advisory group, which includes the United States and Israel.

But the view from Washington is that not all terrorist scenarios have been assessed.

U.S. officials say Greek planners had not considered the possibility snipers might use hills near event sites to fire upon athletes or spectators. Questions also have been raised about whether Greece is ready to handle large-scale casualties if a biological or chemical attack occurs.

The security price tag for Athens -- the first summer Olympics since the Sept. 11, 2001 attacks -- has reached an unprecedented $775 million.

"We're paying for Sept. 11," Athens Mayor Dora Bakoyianni said recently.

Government spokesman Christos Protopapas would not provide details Wednesday on Mueller's trip.

"There will be discussions that concern issues of security that we are jointly working on," Protopapas said. "We are working on issues that concern the international community, such as terrorism. We are cooperating on security for the Olympics."

Greek officials, however, do hope Mueller's trip will help silence critics, mostly in the United States, who accuse Greece of failing to fully stamp out domestic terrorism.

Until the arrest last year of 19 suspected members of November 17, Greece's deadliest terrorist group, Greece had been the focus of numerous international complaints. Most focused on its inability to make any headway against the Marxist-nationalist group, which had killed 23 people in 28 years -- including four American officials.

When Mueller's predecessor, Louis Freeh, visited Greece in late 1998, the entire trip was devoted to Greece's failure to break up November 17.

The issue still generates tension.

Greek officials were angered last week when the U.S. State Department again listed November 17 in its annual report on global terrorism. For Greek authorities, last year's arrests were evidence that it could provide a safe environment for the Olympics.

On Friday, Mueller is expected to present awards for success against November 17 -- which takes it name from the day in 1973 when military dictators crushed a student-led uprising.

"We will have something which will answer to this [State Department listing] in the most definitive way and it won't be something we did," Public Order Minister Giorgos Floridis told parliament.