No, this is not a power that is given to the house and it was setup so that the house, senate and president all need to agree with the exception that if the president does exercise veto power it can be overridden with a 2/3rd vote in both the house and senate. Something to remember, though not directly related, is that in the 1990's congress tried to give the president the line item veto but that was shot down by the supreme court. They said it was unconstitutional because it violated the Presentment Clause.
There is also an important part you are missing about amending the constitution. In order for that to happen from congress it would need a 2/3rd vote in both the house and senate in order for it to get the processes started. Once it passes that route it would need to be ratified by 3/4th of the states which makes it highly unlikely that a single party could force it through especially given how important that section of the constitution is.
Presidents of the United States have repeatedly asked the Congress to give them line-item veto power. According to Louis Fisher in The Politics of Shared Power, Ronald Reagan said to Congress in his 1986 State of the Union address, "Tonight I ask you to give me what forty-three governors have: Give me a line-item veto this year. Give me the authority to veto waste, and I'll take the responsibility, I'll make the cut, I'll take the heat." Bill Clinton echoed the request in his State of the Union address in 1995. Congress attempted to grant this power to the president by the Line Item Veto Act of 1996 to control "pork barrel spending", but in 1998 the U.S. Supreme Court ruled the act to be unconstitutional in a 6–3 decision in Clinton v. City of New York. The court found that exercise of the line-item veto is tantamount to a unilateral amendment or repeal by the executive of only parts of statutes authorizing federal spending, and therefore violated the Presentment Clause of the United States Constitution. Thus a federal line-item veto, at least in this particular formulation, would only be possible through a constitutional amendment. Prior to that ruling, President Clinton applied the line-item veto to the federal budget 82 times
Every Bill which shall have passed the House of Representatives and the Senate, shall, before it become a Law, be presented to the President of the United States: If he approve he shall sign it, but if not he shall return it, with his Objections to that House in which it shall have originated, who shall enter the Objections at large on their Journal, and proceed to reconsider it. If after such Reconsideration two thirds of that House shall agree to pass the Bill, it shall be sent, together with the Objections, to the other House, by which it shall likewise be reconsidered, and if approved by two thirds of that House, it shall become a Law. But in all such Cases the Votes of both Houses shall be determined by Yeas and Nays, and the Names of the Persons voting for and against the Bill shall be entered on the Journal of each House respectively. If any Bill shall not be returned by the President within ten Days (Sundays excepted) after it shall have been presented to him, the Same shall be a Law, in like Manner as if he had signed it, unless the Congress by their Adjournment prevent its Return, in which Case it shall not be a Law. Every Order, Resolution, or Vote to which the Concurrence of the Senate and House of Representatives may be necessary (except on a question of Adjournment) shall be presented to the President of the United States; and before the Same shall take Effect, shall be approved by him, or being disapproved by him, shall be repassed by two thirds of the Senate and House of Representatives, according to the Rules and Limitations prescribed in the Case of a Bill.