France and Italy have signaled their desire to push for a reform of the Schengen framework for border control. One of the most important proposals is the procedural right to temporarily re-establish border controls between two countries. The European Commission is scheduled to present its own plans for amending the Schengen rules next week (4 May).
The Schengen border security legal framework is now part of the EU acquis. Any revision of the Schengen framework goes through a codecision procedure, where the European Parliament is a co-legislator with the Council (see art. 77, para. 2 TFEU). More, the Commission is the only body that can propose legislation on border checks, asylum and immigration (see a contrario art. 76 TFEU). Whatever France and Italy propose is of no relevance; the Member States do not have a right of initiative on these matters.
On all these accounts I am quite skeptical that Italy and France will succeed to push an amendment of the Schengen framework that seriously undermines the principles of the current regime. Any significant policy overhaul must be accompanied by a careful impact assessment and discussions not only among governments of Member States, but also with relevant stakeholders. It will take more than a bilateral summit to do that.