The source told Daily Star newspaper that the clashes erupted in the Mallouleh neighborhood of the city in early hours of Tuesday.
Two soldiers, three gunmen and a Syrian man who was standing on the balcony of his apartment were wounded in the gunfire.
The soldiers were identified as Bilal Yasser Khalil and Ramez Marwan Midlij.
Tensions are running high in Tripoli, a city of nearly 200,000 people and Lebanon's second largest, since the beginning of unrest in the neighboring Syria.
The city has witnessed deadly clashes between supporters and opponents of Syrian President Bashar al-Assad over the past two years.
BA/BA