The Legislative branch.
Suppose it does?
The president can keep the bill "in his pocket" until it is too late to be dealt with in the current legislative session. If he vetoes it instead, Congress can override it and make it into law without introducing new legislation.
A pocket veto will be possible for the president only if Congress adjourns before the president has ten days to sign or veto the bill. If Congress adjourns during this period, the bill does not become law and is effectively vetoed.
The 'pocket veto' only works if Congress is within 10 days of adjourning.
The number of the bill has the initials HR (House Resolution), indicating that it originated in the House of Representatives.
pocket veto
congress is about to adjourn
it becomes law
A bill that originated in the Senate goes to the House of Representatives only if the Senate passes it.
If the President signs a bill that has been passed by Congress, it does not die, but becomes law. I have never heard of a President signing a bill before it was passed by Congress, but if he did and it died, I suppose it would mean that he was in danger of being known as a weak President.
Bills can be passed into law only by the congress; the president can not create any bills. The President could write a proposed bill and get some member of Congress to introduce it and if this happens, you might say the president originated a bill. However, no bill of any importance goes through Congress without substantial changes and usually the House and Senate produce different versions of the bill which must be rectified into one. If the president said before hand that he was going to push certain legislation and if he makes a strong effort to twist arms and spend political capital to get it passed, you might give him credit for originating it.
If he does not approve, he must return the bill, unsigned, within ten days, excluding Sundays, to the house of the United States Congress in which it originated, while the Congress is in session.