WASHINGTON (Reuters) - The United States on Tuesday offered multimillion-dollar rewards for two officials of the Iran-backed Lebanese group Hezbollah as the Trump administration prepared to unveil a strategy to counter Tehran s growing regional influence. Washington will pay up to $7 million for information leading to the arrest of Talal Hamiyah, head of Hezbollah s foreign operations, and up to $5 million for Fuad Shukr, a top Hezbollah military operative, the State Department said. The rewards are the first offered by the United States for Hezbollah operatives in a decade, Nathan Sales, the U.S. counterterrorism coordinator, told reporters.  Today s rewards are another step to increase the pressure on them and their organization,  said Sales. Other extremists for whom the United States is offering rewards include Abu Bakr al-Baghdadi, the head of Islamic State, and Mohammad Jolani, the commander of al Qaeda s Syrian branch. Hamiyah has been on the department s foreign terrorist list since 2015 and Shukr was added in 2013. The United States named Hezbollah as a foreign terrorist organization in 1997. Nicholas Rasmussen, the head of the National CounterTerrorism Center, blamed the group for a litany of attacks around the world, and said it maintains a presence in  nearly every corner of the globe.  Pointing to the arrests of two men in the United States in June for alleged activities on Hezbollah s behalf, Rasmussen said that U.S. intelligence agencies assess that the group is seeking an ability to strike inside  the homeland.   Sales signaled that as part of U.S. President Donald Trump s soon-to-be unveiled Iran strategy, Washington would press countries that have yet to designate Hezbollah as an international terrorist group to do so.  Additionally, some countries have chosen to designate only Hezbollah s military wing, leaving its so-called political wing untouched,  he said, apparently referring to the 28-member European Union.  But that is a false distinction. Make no mistake. Hezbollah has no political wing. It is a single organization, a terrorist organization, and it is rotten to its core.  Designating the group as a terrorist organization is  not merely symbolic,  he continued. By not doing so, he said, countries  limit other governments  ability to freeze Hezbollah s assets, to shut down its front companies, to eliminate its fund-raising and recruiting capabilities and to prosecute Hezbolah associated networks. The United States will need allies in this fight.  But winning support for an intensified campaign against Hezbollah could prove difficult for the administration. The powerful Iran-backed organization is part of Lebanon s fragile coalition government and commands enormous support for the social services it provides. 