David Cameron is being urged to respond to a remain victory in the EU referendum by telling his fellow European leaders they must address immigration and the free movement of citizens if the popularity of the EU is not to plummet to unsustainable levels.

The call to look again at the detail of EU-wide free movement would also be a means for Cameron to acknowledge the depth of public disquiet in Britain over immigration that has been displayed throughout the referendum campaign.

Pro-EU ministers are determined to ensure the European commission does not take the wrong lessons from a remain victory by seeing it as an endorsement of the status quo. They believe Cameron will have a unique mandate to urge EU leaders to listen to their electorates, or face the same “near-death experience” as the UK.

Conservative party whips are also consulting MPs on whether a minister for Europe should be appointed to cabinet as part of a reshuffle that would occur after the referendum, probably in the autumn. The appointment would be a sign that Cameron is determined to engage with the European agenda with new intensity ahead of the UK presidency of the EU in 2017. The presidency provides a chance for the UK to shape the European agenda and is likely to focus on competitiveness.

The current longstanding Europe minister, David Lidington, serves as a minister of state and is responsible for relations with the EU and Russia.

The prime minister will have to return to the EU in the aftermath of the referendum to complete the details of his February deal, including unresolved details about the level of social security benefits and the length of time they can be withheld from people moving to the UK from the EU.

Cameron has tried to raise wider discussions of free movement with his fellow EU leaders, but was rebuffed mainly by Germany, which insisted free movement was an absolute sacrosanct principle.

Some in Downing Street believe that after succeeding in a referendum, he would have the authority to reopen the discussion, and urge a debate on reform. Both Germany and France face elections in 2017, and both campaigns are likely to be dominated by possible EU treaty reform.

The prospect of a new debate on free movement was raised on Wednesday by Simon Fraser, the former permanent secretary at the foreign office, who said: “If we do vote to stay in, is there a realistic prospect in the period afterwards that we could engage with other member states in a discussion about how we manage the principle of freedom of movement without undermining that principle as part of the single market? It does seem to me not only in this country, but elsewhere in Europe this is a a highly sensitive issue.

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“If we were able to have that discussion and debate that might be a way of easing the pressure off this very sensitive question.”

Speaking at a seminar organised by the European Council on Foreign Relations, he said: “I am talking more broadly about the management of the union as it now stands. One of things people are saying to me at the moment is not only the number of people coming to this country from the enlargement states of 2004 but also the number of people who are now coming in to find jobs because of youth unemployment in longstanding members of the EU.”

The Labour leader, Jeremy Corbyn, is opposed to any fresh immigration restrictions, but the former shadow chancellor Ed Balls is one of many party figures calling for a debate on restricting the movement of low-skilled labour within the EU.

The former shadow education minister Tristram Hunt said: “I do think after the referendum it might be time for an area of British leadership to open up this discussion and see what options are available. If we don’t talk about immigration and if we don’t address it, and continue to say there is only one way and there are no other political solutions, then the debate goes in far more reactionary directions.”

The former Polish foreign minister Radosław Sikorski also urged the UK to take the diplomatic initiative in the wake of any vote to remain. He said: “You are not the most important member of the club committee if the club is one that you might leave. So British influence has not been what it has been.”

He added: “The bill is becoming due for decades of Eurosceptic brainwashing in sections of the press.”