Michel Barnier, who is touring Europe to meet officials and business groups, told an audience that such a deal would take "much more time" than the current phase of negotiations, according to sources inside the meetings who spoke to Politico Europe.
According to the Financial Times, Mrs May will reveal the UK Government is prepared to pay €20bn (£17.7bn) to meet its financial obligations to the European Union after 2019.
"How do you build a future relationship if there's no trust, if you haven't honored your commitments?"
"I am asking myself questions".
May has been largely quiet on Brexit since her Conservative Party lost its parliamentary majority in a June election that she need not have called, saying little beyond expressing her support for "a deep and special partnership" with the bloc once Britain leaves in March 2019. She may also offer to make financial contributions to Brussels during a transition period, in order to plug a hole in the so-called divorce bill.
"I am waiting for a proposal on the legal rights of European Union citizens in Britain", said Barnier, who was speaking in French through an Italian translator.
"We are a few days away from the fourth round of negotiations", Barnier said.
"We are not going to mix up models", he is said to have told his audience at the private meetings, "but each model is available".
The EU has insisted that talks can not move onto the issue of future trade relations until "sufficient progress" is made on the aforementioned topics.
Tension between the pair appeared to reach a new high following the most recent round of talks.