The committee that does this in the House of Representatives is called the House Rules Committee. There is no similar committee in the Senate. This is one of the things that gives the majority party in the House so much more power than the majority party in the Senate has.
The majority party in the House controls the Rules Committee. This means that it can set the rules on what amendments, if any, can be offered to bills. It also sets the rules on how long a bill can be debated. By doing things like this, it can push a bill through the House very quickly in a way that is impossible in the Senate.
The presence of the House Rules Committee is one major difference between the House and the Senate.