Authorities issued an urgent warning residents not to use various main roads because of shooting and blockades.
Social media users posted pictures of burning buses. It was not immediately clear if there were casualties.
Local media said the violence was due to the arrest of a drug cartel leader.
The name El Gafe has started circulating in the local media, though BBC Mexico correspondent Katy Watson says little is known about El Gafe - even his real name.
He is thought to have risen to the top of the Gulf Cartel after the capture and deaths of other leaders.
The US consulate in the city of Matamoros, which oversees consular affairs in Reynosa, urged Americans to "use extreme caution and remain in doors".
The Gulf Cartel, the drug cartel dominant in Reynosa and the state of Tamaulipas for years, has been weakened recently by the arrest of several leaders including Mario Armando Ramirez Trevino, known as X-20, in August 2013.
That has led to infighting between different factions in the cartel.
Reynosa and other cities in the state have also been affected by fighting between the Gulf Cartel and their bitter rivals, Los Zetas.
Both cartels have been fighting for control of the region, regarded as a strategic commercial entry point into the US.
Reynosa sits across from the Texan city of McAllen. Together they form a metropolitan area of about two million residents.
Gun battles lasting for several hours have broken out on the streets of Reynosa, a Mexican city on the border with the US that has been plagued by drug cartel violence.