The message comes as the latest round of talks between the two sides entered their third day on Wednesday.
Tajani, an Italian MEP, was speaking after a meeting earlier this week with European Commission Brexit chief negotiator, Michel Barnier.
Tajani, a former Commissioner, said, "With the clock steadily ticking away there are millions of students, workers and families - some of whom have spent their entire lives in another EU country - still facing as much uncertainty now as they did when the British government formally triggered Article 50 in March.
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"A potential agreement will have to be approved by Parliament and our priorities have been consistently clear from the outset: defending the rights of EU citizens, preserving the achievements of the Good Friday agreement and ensuring that the UK fully honours its financial commitments."
The EPP group deputy added, "The British government must come forward with clear positions in order for talks to advance - solving these issues will be essential in securing a deal that will be the building block of a successful future partnership with Britain.
"I will continue to work closely with Barnier, who has the full support of Parliament. In our meeting, he once again impressed me with the preparation that he and his team are displaying in this negotiation. I would urge his counterparts to move forward as time is of the essence and citizens on both sides of the Channel are counting on us to deliver."
The talks this week have been marked by a degree of rancour, with each side making accusations against the other.
The EU insists the UK divorce bill, citizens' rights and the Irish border issue must be resolved before the talks progress to the UK's future trade relations with Brussels.