The Brexit campaign’s pledge that leaving the European Union would mean more money for the health service must be honoured or voters will lose trust in politics, the head of NHS England will warn.

Simon Stevens will cite the leave campaign’s controversial claim that Brexit will bring £350m back under British control to spend on the NHS to argue for more funding for the health service.

He will not call specifically for the figure emblazoned on the side of the Vote Leave bus, which carried Brexiters including Boris Johnson around the country during the referendum.

However, he will insist trust in democracy “will not be strengthened” if the chancellor, Philip Hammond, argues in his budget this month that economic turbulence caused by Brexit means he cannot promise extra cash for the NHS.

Speaking at the NHS Providers conference in Birmingham, Stevens is expected to say: “The NHS wasn’t on the ballot paper, but it was on the battlebus. Vote Leave for a better funded health service – £350m a week.”

He will quote analysis by the Vote Leave campaign director, Dominic Cummings, that Britain would have voted to remain in the EU without the pledge, adding: “Rather than our criticising these clear Brexit funding commitments to NHS patients – promises entered into by cabinet ministers and by MPs – the public want to see them honoured.

“By the end of the NHS’s next financial year – March 2019 – the United Kingdom will have left the European Union. Trust in democratic politics will not be strengthened if anyone now tries to argue: ‘You voted Brexit, partly for a better funded health service. But precisely because of Brexit, you now can’t have one.’

“A modern NHS is itself part of the practical answer to the deep social concerns that gave rise to Brexit. At a time of national division, an NHS that brings us together. An institution that tops the list of what people say makes them proudest to be British. Ahead of the army, the monarchy or the BBC. Unifying young and old, town and country, the struggling and the better off.”

Stevens’ speech will immediately follow a keynote address from the health secretary, Jeremy Hunt, and comes as leading health thinktanks warn that NHS funding will be at one of the lowest rates in its history next year.

The Health Foundation, the King’s Fund and the Nuffield Trust calculated that the NHS needs £4bn more next year to prevent patient care from deteriorating following a joint analysis of NHS finances in England.