Graham warned that efforts to block U.S. efforts to interrogate Tunisian national Ali ani al-Harzi could have dire consequences on future relations between Washington and Tunis.
An alleged member of al Qaeda's North African cell, al-Harzi was temporarily detained by Turkish officials before being extradited to Tunisia, where he is now awaiting trial on terrorism charges.
U.S. intelligence officials, along with members of the FBI, have been in negotiations with the Tunisian government to gain access to al-Harzi or have the suspect transferred to the American detention facility in Guantánamo Bay, Cuba.
So far, both sides have yet to reach an agreement on the suspect's fate.
Al-Harzi has been identified by American authorities as a suspect in the consulate raid in Benghazi, Libya. The attack, carried out by Libyan militants in the region, resulted in the deaths of U.S. Ambassador Christopher Stevens and three other Americans.
Al-Harzi was reportedly en route to Syria, likely to participate in the civil war between anti-government rebels looking to overthrow longtime President Bashar Assad, at the time of his arrest, according to recent reports.
American intelligence officials have also identified Ahmed Abu Khattala, head of the Islamic Libyan militia Ansar al-Sharia, as a suspect in the Benghazi strike.
Abu Khattala, head of the militant group that initially claimed responsibility for the attack, admitted he was at the Benghazi site, but denied playing any role in the attack.
U.S. military and intelligence officials are in the midst of an large-scale inquiry into the events that led up to the Benghazi assault.
A team of FBI investigators recently completed its visit to the consulate site, gathering evidence on the attack and working to identify the perpetrators.
A crucial part of that investigation is gaining access to al-Harzi and finding out what exactly his role was in the violent assault, according to Graham.
"As with any investigation time is of the essence and every day that goes by is a lost opportunity," Graham said.
Tunisia's cooperation, or lack thereof, in the investigation could factor into how the United States deals with the North African nation in years to come.
Washington has been trying to strengthen its diplomatic and military ties with Tunisia's newly democratic government in the wake of last year's Arab Spring movement.
But its reluctance to hand over al-Harzi could damage or sever those burgeoning ties, Graham warned.
"I appreciate the relations between our two nations and remain very hopeful that Tunisia will continue developing into a success story," the South Carolina Republican said.
Graham is one of several GOP lawmakers who have repeatedly hammered the White House, Pentagon and intelligence community over its handling of the Benghazi attack.
Lawmakers in Congress have sent numerous letters and opened up a half-dozen investigations into incident, calling in administration officials to testify at two congressional hearings on the attack.
Republicans are attempting to use the attack to weigh down President Obama's reelection chances in November, highlighting the White House's shifting explanations on what happened in Benghazi.
On Sunday, Sen. John McCain (R-Ariz.) accused the White House of either a “massive cover-up” or “incompetence" in its efforts on the Libyan raid.