Rishi Sunak will plead with other world leaders at a summit on Tuesday for cooperation to tackle illegal migration as he seeks to reassert his authority over the restless right of his party. He will also seek to use a rare meeting with the president of the European court of human rights (ECHR) to win backing for UK attempts to overcome rules that blocked the first scheduled deportation flight to Rwanda. The international system for policing human trafficking is not working, Sunak will say in an address in Reykjavík at a meeting of the Council of Europe – a gathering of leaders from the EU, other European states and the ECHR. Sunak’s renewed push for a Europe-wide approach comes after France previously rejected Britain’s calls for a bilateral returns agreement for migrants crossing the Channel, insisting that there was a need for a wider EU agreement. However, his most important meeting in Iceland could be his discussions with Síofra O’Leary, the ECHR’s president, over planned changes to how rule 39 works – an order that prevented the inaugural deportation flight taking asylum seekers to Rwanda last summer. He is expected to tell her that reform is needed to create an international system that allows states to protect their borders and help people most in need. Looming over the talks is the memory of Sunak caving in last month to demands from hard-right MPs to allow the UK to ignore rulings from the European court of human rights on small boat crossings. Backbench rebels had been pushing the prime minister to harden the illegal migration bill so ministers could ignore interim rulings. One of the Strasbourg court’s rule 39 injunctions blocked the government’s first attempt to deport asylum seekers to Rwanda last year. The trip comes as migration tops the political agenda, with the home secretary, Suella Braverman, arguing in a speech at Monday’s National Conservatism conference that Britain “must not lose sight of the importance of controlling legal migration”, as well as preventing people from entering via unauthorised channels. The Daily Telegraph reported that the Home Office had privately shared figures with No 10 suggesting more than 1.1 million foreign workers and students could arrive in Britain in 2024-25, just as the Tories face a general election test. Speaking before the trip, the prime minister said: “Every single point on each route used by people traffickers to smuggle people across our continent represents another community struggling to deal with the human cost of this barbaric enterprise.” “It is very clear that our current international system is not working, and our communities and the world’s most vulnerable people are paying the price. “We need to do more to cooperate across borders and across jurisdictions to end illegal migration and stop the boats. I am clear that as an active European nation with a proud history helping those in need, the UK will be at the heart of this.”