The U.S. constitution split out foreign relations power to the executive and the legislative branches. Some powers are exclusive of the president like the command of the army and others, like the regulation of the foreign commerce are exclusive to the congress.
In a democracy the president don´t hold absolute power, he is bounded by the rule of the supreme law, the constitution. Article II of it provides legal authority to the president for some things, he has the power to make treaties and appoint ambassadors subject the aprobal of the Senate. He also has the power to recognize new foreign countries and can make executive agreements with other governments with no need of the Senate approval.
The article II also gives the president the power to recommend new legislation to the Congress as well as the power of veto to legislation that he thinks could be against national interest.
Nevertheless this powers, the U.S. constitution on it´s first article gives the Congress some attributes regard the foreign policy including: regulate commerce with foreign nations, declare war, raise and support armies and it gives the final decision to some of the executive foreign powers like the approval of the designation to the ambassadors and the treaties.
Now detailing more on the question itself, the congress plays an oversight role. The annual appropiations allows congretional committees to review the budgets and programs of the bureaucracies, they also get to decide how spend that money. Finally it has the authority to create, eliminate or restructure branch agencies.
I hope this answer help you. Greetings
