The military, which has reported 15 soldiers killed in clashes, has made no comment.
They have demanded a military withdrawal as a condition for talks with the government.
The Abu Sayyaf rebels also said they had captured 13 soldiers and beheaded three of them.
The military, which has reported 15 soldiers killed in clashes, has made no comment.
Last Sunday, the bodies of two resort workers hacked to death were found - one had been beheaded.
A rebel spokesman said Christian missionary Martin Burnham, 41, from Kansas, suffered shrapnel wounds from a grenade during clashes with the military on Basilan island on Monday.
Abu Sayyaf rebels snatched Mr Burnham and his wife, along with 18 others, from a luxury resort off the southern island of Palawan on 27 May.
"He took many hits in the back... He is now in a stable condition, nothing to worry [about]," rebel spokesman Abu Sabaya said in a call to a local radio station.
He said the other hostages were unharmed and were being treated well.
"They wanted to drink Coke, so we went into town and gave them Coke. If we had wanted to kill all of them, we could've easily killed them all," he said.
No martial law
On Wednesday, Philippine President Gloria Arroyo ruled out martial law or emergency powers on Basilan island as options to crush the rebels, her spokesman said.
The Philippines military has been pressing for special emergency powers to deal with the hostage crisis.
Abu Sabaya said negotiations with a government representative had begun with the rebel demand that the military operation end.
"If they wanted to seriously talk to us, they should show it by removing the military," he told Radio Mindanao Network. "While the soldiers are here, there will be no talking."
"The government wants our unconditional surrender. What are they, crazy? If you think you can follow us into the mountains and finish us off, you are wrong," he said.
National Police Chief Leandro Mendoza confirmed that three representatives from the US Federal Bureau of Investigation, who were experts in hostage negotiation, arrived on Wednesday to assist the government.
The group says it wants a separate Islamic homeland in the southern Philippines, though correspondents say its main activity appears to be kidnap for ransom.
Elsewhere in the Philippines, a United States Navy sailor is reported missing after an ambush by suspected communist guerrillas on the slopes of a volcano north of the capital, Manila.