In the ordinary legislative procedure the commission submits a propasal for regulation, which has to be approved by the European Parliament and the Council of Ministers (Council of the Euroean Union, not to be confused with the European Council, which is made up of the heads of state).
This process often needs compromises between these three institutions. Therefore, deliberations are often held in a process called a Trilogue. In the Trilogue representatives of the Commission, Council and Parliament hold informal meetings to come to a compromise regarding regulation, which can be approved by Council and Parliament. The Commission is also party to the Trilogue, but mostly a moderator between Council and Parliament. In practice the Trilogue often preceeds the formal submital of legislative proposals to the European Parliament by the Commission. So by the first reading a compromise has already been established between Parliament and Council.
National governments are involved in this process in the way that the Council is made up of ministers, which are part of the government of each member state. Through these ministers the national governments influence the process and the deliberations in the Trilogue. The national legislature does not have any direct influence in the Trilogue. It only plays an indirect role insofar that it controls the national government.