Does a senator have any jurisdiction over FBI? I would imagine that would be abuse of power.
No they don't...and yes it is.
Senators have no direct control over the FBI which operates as part of the Department of Justice
The Federal Bureau of Investigation (FBI) is the domestic intelligence and security service of the United States, and its principal federal law enforcement agency. Operating under the jurisdiction of the United States Department of Justice, the FBI is also a member of the U.S. Intelligence Community and reports to both the Attorney General and the Director of National Intelligence
However, it is possibly (certainly in fiction) subject to political influence as its budget and oversight can, and often are, overseen by Senate committees and so a senator can have a disproportionate amount of influence.
Essentially, if a senator chooses to ask the FBI for a "favour" there's a strong possibility it will be considered favourably.
The script has this line
Who he called, I don't know. Someone
higher up than anyone in that room with
you. Maybe Representative Vollmer, who
Mason may not own, but does rent from
time to time.
What we find from the novel is that this Representative is actually a Congressman and, more important, does have influence on the Justice Department.
Old Verger also left Mason a U.S. congressman and a member of the House Judiciary Oversight Committee who just couldn’t make ends meet without him.
Hannibal - Thomas Harris
The U.S. House Committee on the Judiciary, also called the House Judiciary Committee, is a standing committee of the United States House of Representatives. It is charged with overseeing the administration of justice within the federal courts, administrative agencies and Federal law enforcement entities