President Francois Hollande told British Prime Minister Theresa May that talks must settle the UK's financial bills and the rights of citizens.
Other pro-EU politicians meeting in Malta expressed a similar view, saying they would not allow "cherry-picking".
Mrs May called for simultaneous talks during exit negotiations.
But in a phone call, Mr Hollande warned her that the withdrawal agreement should come first, a common line among European leaders.
"First we must begin discussions on the modalities of the withdrawal, especially on the rights of citizens and the obligations arising from the commitments that the United Kingdom has made," he said.
Meeting in Malta, members of the centre-right European People's Party (EPP), the largest group in the European parliament, said they would focus on:
The party chairman Manfred Weber said they would "not allow the cherry-picking we saw over the past few decades", and that the UK could not shape the future relationship with the bloc in the way it had done as a member of the EU.
Speaking at the meeting, German Chancellor Angela Merkel did not mention Brexit directly, but said the EU must guarantee the prosperity and security of its citizens or risk them turning away.
She had already said that Britain's commitments to the EU had to be dismantled before talks could move on.
Mrs Merkel ended her speech with a quote from the philosopher Erich Fromm: "He said one loves what one makes a lot of effort for and what one makes a lot of effort for one loves.
"And it's important we must make an effort for this Europe because we love it and we want to leave it behind in a good position for future generations."
The European Council President, Donald Tusk, is expected to set on Friday draft guidelines that will form the basis for EU negotiations with the UK.
They will be amended and then approved by the remaining 27 EU leaders in a month's time, and they are likely to set out a clear negotiation schedule, the BBC's Chris Morris in Malta says.

Bayley, who had pleaded guilty, was sentenced to life imprisonment with a minimum jail term of 35 years.
Ms Meagher, originally from County Louth, was attacked in Melbourne last September as she walked home.
The judge described Bayley as a "recidivist sex offender". Bayley was on parole for previous rape convictions when he attacked Ms Meagher.
The case has led to criticism of the parole system in Australia's Victoria state.
In an earlier hearing, the court was told that Bayley had a history of violent sex attacks.
Judge Geoffrey Nettle said Bayley's rape of Ms Meagher was "savage and degrading".
"This is a recidivist, violent sex offender who intended to kill and this killing is amongst the worst of its kind," he said.
However, he said that Bayley's guilty plea had to be taken into consideration in the sentence.
"As bad as your crimes are, you will have the opportunity in jail to strive for rehabilitation and I propose to set a non-parole period as an incentive for you to try."
In a statement, Ms Meagher's family said: "This despicable crime against Gillian has happened and it has taken nine months to get resolved and to get justice."
"The police and prosecutors have ensured that justice has been done and we as a family want to thank them for this."
Ms Meagher's husband, Tom, said in a statement last week that his life would "never be normal again".
"I miss her insight, fun and wit, her huge smile and infectious personality," he said.
Ms Meagher, who was originally from Drogheda, moved to Australia from Ireland in 2009 with her husband.
She worked for ABC Radio but went missing during a night out with colleagues.
Her body was discovered six days later buried on the outskirts of the city.
News of her death prompted thousands to take part in a rally against violence against women.
France has insisted that the UK must first sort out the terms of its exit from the European Union before talks can move on to future relationships.
[NEXT_CONCEPT]
Australian Adrian Bayley has been jailed for the rape and murder of Irish woman Jill Meagher.