* EU says Brexit schedule under pressure
Sterling weakens on Barnier's warning
* Issues over Irish border, citizens' rights, future deal (Adds quotes, detail)
By Gabriela Baczynska and Robert-Jan Bartunek
BRUSSELS, Feb 9 (Reuters) - A post-Brexit transition is "not a given", Michel Barnier warned Britain on Friday, saying London had "substantial" objections to the European Union's offer and that parts of it were not up for negotiation.
The EU negotiator's stark message, which weakened sterling, came after EU diplomats and officials warned in recent days that sticking points in the talks were threatening the whole Brexit schedule.
Speaking to journalists in Brussels after the latest round of Brexit talks, Barnier listed Britain's issues with the status-quo transition the EU has offered until the end of 2020.
It envisages London will remain bound by all EU laws and pay into the bloc's budget but have no say in decisions. London has said it wants to reach a deal on transition in March.
"If these differences persist, a transition is not a given," Barnier said. "If these disagreements were to persist, there will undoubtedly be a problem."
He said Britain rejects giving lifetime rights to EU citizens who arrive after Brexit but before 2021, demands mechanisms to avoid any new EU laws it dislikes, and opposes a clause to unilaterally suspend Britain's access to the single market in case of disputes.
"Frankly, I am surprised by these disagreements," he said. "In demanding the benefits of the single market, the customs union ... the United Kingdom should logically accept all the rules and obligations until the end of the transition."
Barnier stressed the EU was waiting for London to explain what sort of future relationship it wanted with the EU and how to avoid border controls between the north and south of Ireland if it leaves the bloc's customs union in the future, as Prime Minister Theresa May has said it will.
"The sooner the UK makes its choices, the better," Barnier said. "A UK decision to leave the single market and leave the customs union would make border checks unavoidable."
"We focus on solutions to avoid a hard border ... any solution must be precise, clear and unambiguous."
Barnier said Britain had to sort out all issues related to its divorce from the bloc - including ties with the EU nuclear agency Euratom and personal data protection - to win a transition period after it leaves the EU, as is due next March.
"There is no transition possible without a withdrawal agreement," he said.
London particularly dislikes the EU retaining the right to to cut Britain off from its market. Britain's Brexit negotiator David Davis on Friday called its approach "discourteous".
"You will not find in our attitude ... the least trace of discourtesy or willingness to punish," Barnier responded.
He said the next round of Brexit negotiations, expected in two weeks' time, would look again at the transition, as well as whether Britain would be covered by EU deals with third countries during that time.
Britain wants third countries to continue their current treatment of it and has said it will seek the EU's help in persuading them to do so.
The bloc has said it would expect Britain to remain bound by such deals during transition but cannot guarantee third parties will grant London the same benefits.
The EU is increasingly considering deviations from the Brexit schedule, which assumes the other 27 EU national leaders will in March give Barnier a mandate for talks with Britain on its future ties with the bloc.
"Maybe we can wait more," a senior EU diplomat said. "But if we don't have it all sorted out properly by the Brexit date, there will be a cliff-edge. They will crash out, there will be even more damage." (Additional reporting by Phil Blenkinsop and Alissa de Carbonnel; Writing by Gabriela Baczynska; Editing by Andrew Roche)