You seem to misunderstand the process. Bills originate in congress, and the president serves as a gatekeeper deciding whether to accept bills passed by congress. The president can ask congress to pass a certain provision, but congress is under no obligation to accommodate such requests.
So first, one house passes a bill which has been introduced by one of its members (and probably has been approved by one or more committees before it can be brought to a vote). Second, the other house does the same, also subject to the committee process. Only after both houses of congress have passed a bill is it sent to the president for consideration. The president may sign the bill or reject it.
In the latter case, where there is a disagreement between congress and the president, only congress can overcome the disagreement, by overriding the veto, which requires each house to vote by a supermajority of two thirds or more.