Both the president and Congress try to influence the bureaucracy while the bureaucracy, in turn, tries to influence them.
The president tries to control the bureaucracy through appointing the heads of departments and through his (or someday her) budgeting requests. The Congress tries to control the bureaucracy through hearings and through their role in voting on the budget. In turn, bureaucratic agencies try to influence the elected branches. They try to gain support for their missions so that they will continue to get funded in ways that allow them to do what they see as their job.
Bureaucracy refers to government institutions that are responsible for the implementation of public policies. They are ideally supposed to be free from political accountability and are expected to discharge their mandate autonomously. There are four elements of Federal bureaucracy and they are the Cabinet department, Independent executive agencies, Independent regulatory agencies i.e. EPA, and Government organizations.
The President is charged with the responsibility of choosing appointees for the bureaucratic institutions but the appointees must be approved by the Senate. The President has the power to issue executive orders which influence policies. The President also has the capacity to exert his influence on the budgets catering to the agencies' programs and operations.
Congress, on the other hand, exercises an oversight role on the bureaucracy with regards to appointments, budgets and accountability. Even though Congress lacks the capacity to choose appointees, members can influence the President into making certain favorable choices. Further, Congress has the capacity to change and allocate new responsibilities to different agencies.