The fate of 4.5 million citizens threatens to be one of the thorniest issues in the Brexit talks.
In an interview with AFP, Italy's Tajani said parliament would be a "protagonist" and hoped to hold its casting vote on a Brexit deal in late 2018 or early 2019.
"There is one sole red line: the interest of European citizens. There are no other red lines," Tajani said late Wednesday at his office overlooking Brussels.
The fate of 4.5 million citizens -- both EU nationals in Britain and Britons living in the EU -- threatens to be one of the thorniest issues in the Brexit talks.
Currently on both sides they have the right to live and work in each others' countries, as well as receive benefits in many cases, but that is all at risk.
British Prime Minister Theresa May is set to formally trigger the two-year divorce process from the European Union on March 29, beginning fevered negotiations.
The European Parliament head echoed EU Brexit negotiator Michel Barnier, who said Wednesday that guaranteeing the rights of European citizens was the "absolute priority."
Tajani said talks on a future trade relationship could not start until Britain reaches a divorce agreement, including citizens' rights and Britain's exit bill.
"You have to divorce," said Tajani, who took over from Germany's Martin Schulz as European Parliament head in January.
"Then afterwards you can sort out the relations between the two divorcees who live next door to each other."
The Brexit talks will be led by Barnier and the remaining 27 EU member states, but Tajani insisted the 751-seat parliament would still be a "protagonist".
MEPs will hold a debate on Brexit in Strasbourg in the first week of April followed by a vote on a resolution, he said.
He warned that time would also be short for talks, with a draft deal needing to be in place months before Britain is due to formally leave the EU in March 2019.
"I hope that we will hurry up. The vote in parliament is crucial, we have to vote at the end of 2018, beginning of 2019," he said.
Tajani also warned that parliament could veto any Brexit deal it views as unsatisfactory.
"It's a free vote. Let's hope that parliament votes yes, but we have to see the content, no?" he said.
It was however "too early" to say if he expected a transitional deal in case a trade agreement is not sorted out by the end of the two years, or if parliament would insist that any such deal would be overseen by the European Court of Justice -- itself a red line for London.
"It won't be easy. It will be technical, legal, truly complicated work," he warned.
This weekend Tajani heads to his home city of Rome for a summit to celebrate the 60th anniversary of the signing of the EU's founding treaty.
Tajani said he backed plans for a declaration by the EU leaders in Rome -- meeting without Britain -- to call for a "multi-speed" Europe where countries can choose to push ahead with integration in some areas, while others opt out.
However, he insisted others should not be left behind.
"I am not against two, three or four member states which decide to be in the lead and open a way for the others," he said.
"If France, Germany, Italy and Spain, for example, decide to cooperate on defence, and work together, we will prepare the way."
Meanwhile with Europe still wary about new US President Donald Trump's commitment to transatlantic ties after a series of disparaging comments about the bloc, Tajani said he would very much welcome Trump to speak at the European Parliament when the US leader visits Brussels for a NATO summit in May.
"We'll have to see what he does, personally I would have nothing against it," he said.
"We have to give him a way of understanding the situation we have in Europe. That's not easy."