"The military sources have not given their casualties so far, but we have been told [through sources] that two [soldiers] have been killed."
Civilians are believed to have fled Bajaur amid the fighting, Hyder reported.
"Most of the civilians in Bajaur have fled the region, giving the Pakistani military the upper hand and allowing them to enter the area."
The tribal region is one of the most tense areas inside Pakistan, being a favoured base for fighters linked to the Taliban and al-Qaeda.
"It is extremely volatile in Bajaur ... the Pakistanis and the United States have differences over this tribal region," Hyder said.
"The Pakistanis fear that if the Americans go into this region, they could lose the battle for hearts and minds there."
Concerns over US military involvement in Pakistan come after General Ashfaq Parvez Kayani, Pakistan's military chief, criticised a US air raid that killed civilians inside Pakistan last week.
Kayani said that the raid, which was launched by the US from neighbouring Afghanistan, could cause a backlash in favour of al-Qaeda and the Taliban.
"Falling for short-term gains while ignoring our long-term interest is not the right way forward," he said on Wednesday.
According to the American newspaper The New York Times, George Bush, the US president, "secretly approved" orders to allow US special forces to operate inside Pakistan, without the approval of Islamabad.
But Kayani said that "the sovereignty and territorial integrity of [Pakistan] will be defended at all cost and no external force is allowed to conduct operations inside Pakistan".