The leaders of Britain’s biggest businesses have told Theresa May to urgently rethink her post-Brexit immigration plan, warning it would trigger labour shortages and cause “worrying” damage to the UK economy.

Despite a broadly warm reception for the prime minister and cautious backing for her draft EU withdrawal agreement secured last week, business leaders at the CBI annual conference in London on Monday said it was worrying that May aimed to “pull up the drawbridge” on EU migrant workers.

Carolyn Fairbairn, the CBI’s director general, said that business leaders had “major differences” with the government on immigration: “We don’t disagree with the idea of [taking back] control, it’s the way you do it: pulling up the drawbridge, very quickly as we understand, would be very damaging.

“What is your plan? Where are the people going to come from who are going to work in our businesses and fuel our economy if you go down this route,” she said.

May had earlier used a speech to the CBI conference to sell her Brexit plan to business leaders while taking a tougher line on immigration, arguing that she would secure full control of the UK’s borders and stop EU migrants “jumping the queue” to come to the UK after Brexit.

Romana Abdin, the chief executive of Simplyhealth, said the choice of language used by May was worrying, as health sector workers such as dentists and physiotherapists were now leaving the UK. “When we hear the prime minister talking about quotas and restricting talent, that worries us on behalf of everyone.”

Warren East, the chief executive of Rolls-Royce, said some European workers had already started to leave since the Brexit vote two years ago: “[It’s] not vast numbers, but enough to be ‘oh, this is an issue’.”

Official figures last week showed an unprecedented 132,000 annual drop in workers from the EU, with most of the reduction coming from eastern European people. The trend has alarmed business leaders.

Despite fears about her immigration plans, the atmosphere at the conference was broadly supportive of the prime minister after a tempestuous week of political infighting and resignations.

Business leaders warmly applauded her speech and argued that her deal was better than a no-deal Brexit.

Surinder Arora, the chairman of the Arora Group, which operates the hotel where the conference was being held, told the prime minister he was struggling to find staff for the venue since the EU referendum, although he broadly supported her plan. “We all need to be behind you,” he said. “We’ve done so well getting so far and we can’t please everyone, but let’s just get on.”

Ross McEwan, the chief executive of Royal Bank of Scotland, told the Guardian it was best to “back the prime minister” given the lack of workable alternatives on the table. The chief of the majority government-owned bank, which has made a £100m provision to guard against losses connected to Brexit, said that a no-deal Brexit would damage the economy and that May’s deal would give firms the transition period they required to plan for the final Brexit deal.

Fairbairn said the prime minister’s Brexit deal was required to reduce the risk of further business investments “flooding out” of Britain that would prevent the UK from improving its weak levels of economic productivity. “The prime minister’s agreement is not perfect. It is a compromise. But it is hard-won progress,” she said.