European Parliament Brexit coordinator Guy Verhofstadt's comments came when he appeared before the employment committee on Wednesday.
Addressing the ongoing Brexit negotiations, he expressed his "fear that some people in the UK are still trying to establish a hard Brexit or would even want to walk out of the negotiations altogether."
The ALDE group leader said, "Our continued relationship is too important for our citizens and our companies to be jeopardised by political games."
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Verhofstadt went on, "A divorce is never easy, but a strong future partnership is in the best interest of all."
He was updating MEPs on the state of play and next steps in the Brexit negotiations after the summer break.
The debate focused on what Parliament has said are its priorities - ensuring the rights of EU citizens in the UK, as well as of UK citizens in the EU. Specific issues, such as family reunification, social contributions recognition, mutual recognition of qualifications and frontier workers were also addressed at the meeting.
Verhofstadt pointed out that the Parliament should adopt a resolution in October assessing the progress of the negotiations, emphasising the need to protect citizens' rights and a stand against a hard border between Ireland and the UK, which would be against the Good Friday agreement.
MEPs mostly reaffirmed the Parliament's position and expressed their support to the Brexit steering group, which is chaired by Verhofstadt.
S&D group Chair Gianni Pittella told this website, "The decision to have a resolution is in the hands of the Conference of Presidents and depends on the outcome of the current round of negotiations.
"Should we have a resolution, of course, citizens' rights are an absolute priority for the S&D and the Parliament has a clear and united view on this, so I have no doubts it will be supported by a large majority."
Further reaction came from Terry Reintke, a Greens/EFA member of the employment and social affairs committee.
She told the Parliament Magazine, "The picture of the Brexit negotiations remains messy. But there is no time for distraction: The first important step that will have to be taken jointly by the negotiators is to safeguard citizens' rights. The proposal by the British government is not sufficient to clarify the situation for millions of people who deserve answers."
Elsewhere, Conservative MEP Syed Kamall, co-Chair of the ECR group, was very critical of Verhofstadt, saying, "His role as Brexit coordinator is to act as a link between the European Parliament and the Commission. It is not his job to try and impose a position on MEPs.
"For him to suggest, with more than a week of negotiations still to take place, that sufficient progress will not have been made by October to allow talks to begin on Britain's future relationship with the EU, is premature and way beyond his pay grade."
He added, "Unless he has a crystal ball how can he predict the outcome of negotiations at which he is not even present?
"These are matters for the whole Parliament to debate and Verhofstadt should not try to dictate the outcome. He needs to stop trying to do Michel Barnier's job and stick to his own role as an honest broker."
UKIP's Ray Finch, who was at the meeting, said afterwards,"Verhofstadt claims that more information is required to start the negotiations. It is now 14 months from the referendum, there's been ample time for both sides to get on with this.
"The EU are simply stalling for time in the hope that we give up and don't leave in the end. We are in a position of strength, they need us far more than we need them. Our government would be wise to remind them of that."
The meeting comes after Michel Barnier said this week that he cannot accede to UK demands to be flexible until he knows what Britain wants.
"To be flexible you need two points, our point and their point," Barnier told reporters on the sidelines of more Brexit talks in Brussels on Wednesday. "We need to know their position and then I can be flexible."
His comments are taken as further proof that the third round of talks which started on Monday and will conclude on Thursday have failed to make much progress, with both sides venting their frustrations in public.
Talks kicked off with the EU side rebuking the UK for being unprepared, while Britain continues to butt against the EU's schedule for talks - first the divorce and then the future relationship.
The thorny issue of the bill was tackled during the latest negotiations but the raft of position documents Britain published in the lead-up to talks did not include a paper on how it proposed to calculate the financial settlement of the divorce.