From the US Constitution, Art. I sec. 7
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...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.
The Constitution makes clear that the president clearly "has a say in the matter." The president can choose to veto, sign or do nothing. The president can cause essentially three outcomes: enact the law (by signing the bill or doing nothing for ten days while Congress is in session); veto the bill (by sending it back to Congress, subject to a potential override); or "pocket veto" the bill (by doing nothing for ten days while Congress is not in session, subject to no override). Simply because presidents do not always have the last word does not minimize their power in this situation.
Let's look at the likelihood of potential outcomes. First, we can examine the so-called pocket veto. Because of issues arising from the recess appointments clause, Congress effectively has not adjourned since 2007 by opening a daily session at least once every three days, even if such a session is only "pro forma." Although this is intended to prevent "radical" recess appointments, it also in practice prevents pocket vetos. The second option is that the president enacts the law, in which case nothing more happens. The final option is a veto and then attempted override.
Today, the House of Representatives is controlled by the Democrats with a 235/434 majority (the outcome of one seat is currently disputed), or ~54% of the House. In the Senate, the Republicans have a 53/100 seat majority, or 53% of the Senate. The best case for Congress to override a veto would involve 55 House Republicans (28% of the House Republican caucus) and 20 Senate Republicans (38% of the Senate Republican conference) to defect. Further, the Senate rules would make it very unlikely for a bill to come to a vote if the Senate Republican leadership (Sen. McConnell and some others) are opposed to it.