The French government has called on Boris Johnson to use David Frost’s resignation as Brexit minister to “rebuild trust” with the EU amid uncertainty over the prime minister’s approach in the new year.

Clément Beaune, France’s EU affairs minister, who had a series of run-ins with the prime minister’s pugnacious minister, suggested that Downing Street should use the moment to reset the troubled relationship.

He said: “We had difficult relations but we always continued the dialogue. I send my best with respect to David Frost after his resignation. It is time for the British government to rebuild a climate of trust with France and the EU in the interest of all”.

While Frost has proved to be a tough negotiating partner over the last two and a half years, his formal resignation on Sunday set off alarm bells in Brussels and EU capitals.

In recent weeks, the UK government softened its approach to the post-Brexit arrangements for Northern Ireland, with the two sides brokering a Christmas truce in the talks on a relatively positive note.

There is concern that Johnson, under huge pressure from the right of his Conservative party over Covid restrictions, will feel the need to replace Frost with someone who will want to unravel the progress.

On Sunday, reports suggested that figures such as Iain Duncan Smith, the former work and pensions secretary, known in Westminster as IDS, or even David Davis, who resigned as Theresa May’s Brexit secretary in 2018, could be in the running. Both men have called for the ditching of the Northern Ireland protocol in the past.

Provided by The Guardian Photograph: John Thys/AFP/Getty Images

In reference to Johnson’s apparent sudden changes in position on Brexit over the years, Dominic Cummings, the prime minister’s former chief adviser, tweeted: “IDS for Brexit minister is the Platonic Form of pure [trolley]”.

One EU diplomat said of the minister’s resignation: “Hopefully the new negotiator will be more pragmatic, making good relations with the EU and its member states relations a priority over the pursuit of a pure, antagonistic Brexit - we’re not holding our breath.”

Frost was recognised in Brussels as having both the prime minister’s ear and standing among the most vocal Brexiters in the Tory party. His apparent buy-in to a notable change of tone and policy in recent weeks was seen as a positive.

While the EU has not wholly embraced the change of approach, insisting that Downing Street was still seeking to renege on its past agreements, the UK had been offering to focus on issues around trade friction rather than pursue a more thorough going rewrite of the protocol.

But while there are some concerns about the future, few tears will be shed in Brussels over Frost’s departure from the scene. He had a willingness to push negotiations to the edge, even on the sensitive issue of the Irish border.

Frost inherited and finished the negotiations on the so-called Northern Ireland protocol in the withdrawal agreement, a compromise that guarantees that there is no border on the island of Ireland. But the minister had been scathing about the arrangement, which in effect keeps Northern Ireland in the EU’s single market for goods and draws a customs border down the Irish Sea.

His command paper of July emphasised that this outcome had been forced on the Johnson administration due to errors under the previous prime minister.

Frost sought to unpick much of that deal, with his most contentious demand being that the European court of justice (ECJ) should not be the arbiter of disputes over the implementation of the law in Northern Ireland.

In recent days, there has been a shift to accept that the ECJ will play a role, albeit merely as a reference point on EU law for an independent arbitration panel. Frost privately insisted this was not a new position, but other British officials have indicated that it was a shift, hinting at difficult conversations within government.

Diplomats and officials in Brussels were not generous in their assessment of the minister, known to the prime minister as “the Great Frost”.

“Lord Frost never got Brexit done; he actually made sure it endured using the Northern Ireland protocol to get his way,” one senior EU diplomat said. “But one year on from the deal he negotiated, what did it bring the UK? Apart from mistrust and deteriorated relationships with most member states?”

The diplomat added: “Frost seemed to have a very ideological idea of what Brexit meant and that didn’t make for good neighbourly relations. For now the European court of justice was a bridge too far for Downing Street, but it will depend on his successor whether we indeed see a more pragmatic line emerge.”

A further major issue that will require the attention of Frost’s replacement is fisheries, where the French government continues to insist that it has been hard done by in terms of licences for vessels operating in British waters.

Frost has also voiced his frustration that the UK has not been provided with access to Horizon Europe, the EU’s research and innovation programme. Two months ago, the veteran MP Bill Cash, chair of the Commons European scrutiny committee, claimed British scientists were being frozen out of the £80bn flagship research programme because of the ongoing dispute over Northern Ireland.