The president of the European parliament has pressed Theresa May to agree a swift deal on the future of EU citizens living in Britain during their first one-on-one meeting in Downing Street.

Antonio Tajani, an Italian who recently replaced German veteran Martin Schulz as the international representative of the key EU institution, said he was optimistic the prime minister would respect all existing rights enjoyed by European citizens because she wanted the same in return for British citizens in the rest of Europe.

“I underlined during the meeting [that] for the EU parliament this is most important point,” Tajani told reporters afterwards. “[We must] strongly defend the right of EU citizens living in the UK and have a clear framework in the next months.”

He added: “For us it’s important to ensure that Brexit does not have negative effects on their life and rights they are enjoying.”

Stressing that there should be no diminution of citizens’ rights after Brexit potentially sets a high bar for any reciprocal deal since May is also determined to limit freedom of movement rules enshrined in EU law.

But Tajani described the protection of rights for EU citizens settled in the UK and Britons living in rest of Europe as a “red line” in negotiations.

“We want to defend the same today rights,” he said. “This is very clear in the text approved by a very large majority [by the European parliament]. For us, it is a priority and it is a red line. It’s impossible to go back,” he said.

Rights acquired under EU law include the right of any EU citizen to live, to work, to own a business in any other EU country. It also allows for other family members to join them, a right EU citizens have been campaigning to protect as this accommodates future care of widowed parents.

Despite describing the meeting as good-natured, Tajani said he had so far failed to secure any firm commitment from May to address the European parliament, something MEPs had been calling for to demonstrate their importance in ratifying any Brexit deal.

Nevertheless Tajani welcomed May’s decision to call a snap election, saying it was unlikely to delay Brexit talks by more than a week and would provide long-term clarity.

“To have a new government before the beginning of the negotiation is good, not only for the UK but for us, because we will have the same negotiators, the same prime minister and we will know the real situation in the UK,” said the EU president.

“It is better for us to work with the same government and not with a potential election campaign [looming].”

The EU president struck a more positive note than some MEPs about the prospects for moving on to trade talks with Britain before the end of the two-year window allocated for the exit negotiations.

“We hope before the end of 2017 to have a framework on the exit and immediately afterwards it is possible to start with the negotiations for the day after,” said Tajani, whose press conference was briefly interrupted by a phone call from the EU’s chief Brexit negotiator, Michel Barnier.

“I will inform the [European] council of this meeting and that it was positive. It’s the beginning, there’s a long way but it’s important to start well.”